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NCERT solutions for class 11 maths chapter 9 sequences and series are discussed here. Sequence means the progression of numbers in a definite order and series means the sum of the objects of the sequence. In the previous classes, you have studied about arithmetic progression(A.P). In this NCERT Book chapter, we will discuss more arithmetic progression(A.P) and geometric progression(G.P). In this article, you will get sequences and series class 11 NCERT solutions. these NCERT solutions are prepared by experts in keeping in mind of latest syllabus of CBSE 2023.
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The chapter 9 class 11 maths includes topics such as arithmetic progression(A.P), geometric progression(G.P), arithmetic means(A.M), geometric mean(G.M), the relationship between A.M. and G.M, sum to n terms of special series, sum to n terms of squares and cubes of natural numbers are covered in this NCERT Book chapter. Students can find all NCERT Solutions for Class 11 here and practice them. You will get questions related to these topics in the class 11 maths chapter 9 NCERT solutions. In this ch 9 maths class 11, there are two types of sequence.
Also read:
Progression:
A sequence whose terms follow certain patterns is known as a progression.
Arithmetic Progression (AP):
An arithmetic progression (A.P.) is a sequence where the terms either increase or decrease regularly by the same constant called the common difference (d).
The first term is denoted by a, and the last term of an AP is denoted by l.
The nth term of an AP: an = a + (n – 1)d.
nth term of an AP from the last term: a’n = an – (n – 1)d.
Common difference of an AP: d = an – an-1, for all n > 1.
Sum of n Terms of an AP: Sn = n/2 [2a + (n – 1)d] = n/2 (a1+ an).
A sequence is an AP if the sum of n terms is of the form An2 + Bn, where A and B are constants, and A = half of common difference, i.e., 2A = d.
Arithmetic Mean:
If a, A, and b are in an AP, then A = (a + b)/2 is called the arithmetic mean of a and b.
If a1, a2, a3,…, an are n numbers, then their arithmetic mean is given by: A = (a1 + a2 + a3 + ... + an)/n.
The common difference d is given as: d = (b – a)/(n + 1).
The Sum of n arithmetic means between a and b is n (a + b)/2.
Geometric Progression (GP):
A sequence in which the ratio of two consecutive terms is constant is called a geometric progression (GP).
The constant ratio is called the common ratio (r), i.e., r = an+1/an, for all n > 1.
The general term or nth term of GP: an = ar(n-1).
nth term of a GP from the end: a’n = 1/rn-1, where l is the last term.
If a, b, and c are three consecutive terms of a GP, then b2 = ac.
Geometric Mean (GM):
If a, G, and b are in a GP, then G is called the geometric mean of a and b and is given by G = √(ab).
If a, G1, G2, G3, …, Gn, b are in GP, then G1, G2, G3, …, Gn are in GMs between a and b.
The common ratio r is given as: r = (b/a)(1/n+1).
The GM of a1, a2, a3,…, an is given by: GM = (a1 . a2 . a3 …an)(1/n).
Product of n GMs is G1 × G2 × G3 × … × Gn = Gn = (ab)(n/2).
Sum of First n Natural Numbers:
Sum of the first n natural numbers is:
Σn = 1 + 2 + 3 + … + n = n(n+1)/2.
Sum of Squares of First n Natural Numbers:
Sum of squares of the first n natural numbers is:
Σn2 = 12 + 22 + 32 + … + n2 = n(n+1)(2n+1)/6.
Sum of Cubes of First n Natural Numbers:
Sum of cubes of the first n natural numbers is:
Σn3 = 13 + 23 + 33 + .. + n3 = (n(n+1)(2n+1)/6)2.
Free download NCERT Solutions for Class 11 Maths Chapter 9 Sequences and Series for CBSE Exam.
Sequences and series class 11 questions and answers - Exercise: 9.1
Question:1 Write the first five terms of each of the sequences in Exercises 1 to 6 whose nth terms are:
Answer:
Given :
Therefore, the required number of terms =3, 8, 15, 24, 35
Question:2 Write the first five terms of each of the sequences in Exercises 1 to 6 whose nthterms are:
Answer:
Given :
Therefore, the required number of terms
Question:3 Write the first five terms of each of the sequences in Exercises 1 to 6 whose nthterms are:
Answer:
Given :
Therefore, required number of terms
Question:4 Write the first five terms of each of the sequences in Exercises 1 to 6 whose nth terms are:
Answer:
Given :
Therefore, the required number of terms
Question:5 Write the first five terms of each of the sequences in Exercises 1 to 6 whose nth terms are:
Answer:
Given :
Therefore, the required number of terms
Question:6 Write the first five terms of each of the sequences in Exercises 1 to 6 whose nth terms are:
Answer:
Given :
Therefore, the required number of terms
Question:7 Find the indicated terms in each of the sequences in Exercises 7 to 10 whose nth terms are:
Answer:
Put
Put n=24,
Hence, we have
Question:8 Find the indicated terms in each of the sequences in Exercises 7 to 10 whose nth terms are:
Answer:
Given :
Put n=7,
Heence, we have
Question:9 Find the indicated terms in each of the sequences in Exercises 7 to 10 whose nth terms are:
Answer:
Given :
Put n =9,
The value of
Question:10 Find the indicated terms in each of the sequences in Exercises 7 to 10 whose nth terms are:
Answer:
Given :
Put n=20,
Hence, value of
Answer:
Given :
Hence, five terms of series are
Series
Answer:
Given :
Hence, five terms of series are
Series
Answer:
Given :
Hence, five terms of series are
Series
Question:14 The Fibonacci sequence is defined by
Answer:
Given : The Fibonacci sequence is defined by
Class 11 maths chapter 9 question answer - Exercise: 9.2
Question:1 Find the sum of odd integers from 1 to 2001.
Answer:
Odd integers from 1 to 2001 are
This sequence is an A.P.
Here , first term =a =1
common difference = 2.
We know ,
The , sum of odd integers from 1 to 2001 is 1002001.
Question:2 Find the sum of all natural numbers lying between 100 and 1000, which are multiples of 5.
Answer:
Numbers divisible by 5 from 100 to 1000 are
This sequence is an A.P.
Here , first term =a =105
common difference = 5.
We know ,
The sum of numbers divisible by 5 from 100 to 1000 is 98450.
Answer:
First term =a=2
Let the series be
Sum of first five terms
Sum of next five terms
Given : The sum of the first five terms is one-fourth of the next five terms.
To prove :
L.H.S :
Hence, 20th term is –112.
Question: 4 How many terms of the A.P. are needed to give the sum –25?
Answer:
Given : A.P. =
Given : sum = -25
Answer:
Given : In an A.P., if pth term is 1/q and qth term is 1/p
Subtracting (2) from (1), we get
Putting value of d in equation (1),we get
Hence,the sum of first pq terms is 1/2 (pq +1), where .
Question:6 If the sum of a certain number of terms of the A.P. 25, 22, 19, … is 116. Find the last term.
Answer:
Given : A.P. 25, 22, 19, ….....
a=25 , d = -3
n could not be so n=8.
Last term
The, last term of A.P. is 4.
Question:7 Find the sum to n terms of the A.P., whose term is 5k + 1.
Answer:
Given :
Comparing LHS and RHS , we have
and
Putting value of d,
Question:8 If the sum of n terms of an A.P. is , where p and q are constants, find the common difference
Answer:
If the sum of n terms of an A.P. is ,
Comparing coefficients of on both side , we get
The common difference of AP is 2q.
Answer:
Given: The sums of n terms of two arithmetic progressions are in the ratio.
There are two AP's with first terms = and common difference =
Substituting n=35,we get
Thus, the ratio of the 18th term of AP's is
Answer:
Let first term of AP = a and common difference = d.
Then,
Given :
Now,
Thus, sum of p+q terms of AP is 0.
Question:11 Sum of the first p, q and r terms of an A.P. are a, b and c, respectively. Prove that
Answer:
To prove :
Let and d be the first term and the common difference of AP, respectively.
According to the given information, we have
Subtracting equation (2) from (1), we have
Subtracting equation (3) from (2), we have
Equating values of d, we have
Dividing both sides from pqr, we get
Hence, the given result is proved.
Question:12 The ratio of the sums of m and n terms of an A.P. is . Show that the ratio of mth and nth term is .
Answer:
Let a and b be the first term and common difference of a AP ,respectively.
Given : The ratio of the sums of m and n terms of an A.P. is .
To prove : the ratio of mth and nth term is .
Put , we get
From equation (1) ,we get
Hence proved.
Question:13 If the sum of n terms of an A.P. is and its term is 164, find the value of m.
Answer:
Given : If the sum of n terms of an A.P. is and its term is 164
Let a and d be first term and common difference of a AP ,respectively.
Sum of n terms =
Comparing the coefficients of n on both side , we have
Also ,
m th term is 164.
Hence, the value of m is 27.
Question:14 Insert five numbers between 8 and 26 such that the resulting sequence is an A.P.
Answer:
Let five numbers be A,B,C,D,E.
Then
Here we have,
Thus, we have
Thus, the five numbers are 11,14,17,20,23.
Question:15 If is the A.M. between a and b, then find the value of n.
Answer:
Given : is the A.M. between a and b.
Thus, value of n is 1.
Answer:
Let A,B,C.........M be m numbers.
Then,
Here we have,
Given : the ratio of and numbers is 5 : 9.
Putting value of d from above,
Thus, value of m is 14.
Answer:
The first instalment is of Rs. 100.
If the instalment increase by Rs 5 every month, second instalment is Rs.105.
Then , it forms an AP.
We have ,
Thus, he will pay Rs. 245 in the 30th instalment.
Answer:
The angles of polygon forms AP with common difference of and first term as .
We know that sum of angles of polygon with n sides is
Sides of polygon are 9 or 16.
Class 11 maths chapter 9 question answer - Exercise: 9.3
Question:1 Find the and terms of the G.P.
Answer:
G.P :
first term = a
common ratio =r
the nth term of G.P
Question:2 Find the term of a G.P. whose term is 192 and the common ratio is 2.
Answer:
First term = a
common ratio =r=2
term is 192
is the term of a G.P.
Question:3 The terms of a G.P. are p, q and s, respectively. Show that
Answer:
To prove :
Let first term=a and common ratio = r
Dividing equation 2 by 1, we have
Dividing equation 3 by 2, we have
Equating values of , we have
Hence proved
Question:4 The term of a G.P. is square of its second term, and the first term is -3. Determine its term.
Answer:
First term =a= -3
term of a G.P. is square of its second term
Thus, seventh term is -2187.
Question:5(a) Which term of the following sequences:
Answer:
Given :
n th term is given as 128.
The, 13 th term is 128.
Question:5(b) Which term of the following sequences:
Answer:
Given :
n th term is given as 729.
The, 12 th term is 729.
Question:5(c) Which term of the following sequences:
Answer:
Given :
n th term is given as
Thus, n=9.
Question:6 For what values of x, the numbers are in G.P.?
Answer:
Common ratio=r.
Thus, for ,given numbers will be in GP.
Answer:
geometric progressions is 0.15, 0.015, 0.0015, ... .....
a=0.15 , r = 0.1 , n=20
Question:9 Find the sum to indicated number of terms in each of the geometric progressions in
Answer:
The sum to the indicated number of terms in each of the geometric progressions is:
Question:12 The sum of first three terms of a G.P. is and their product is 1. Find the common ratio and the terms.
Answer:
Given : The sum of first three terms of a G.P. is and their product is 1.
Let three terms be .
Product of 3 terms is 1.
Put value of a in equation 1,
The three terms of AP are .
Question:13 How many terms of G.P. , … are needed to give the sum 120?
Answer:
G.P.= , …............
Sum =120
These terms are GP with a=3 and r=3.
Hence, we have value of n as 4 to get sum of 120.
Answer:
Let GP be
Given : The sum of first three terms of a G.P. is 16
Given : the sum of the next three terms is128.
Dividing equation (2) by (1), we have
Putting value of r =2 in equation 1,we have
Question:15 Given a G.P. with a = 729 and term 64, determine
Answer:
Given a G.P. with a = 729 and term 64.
(Answer)
Question:16 Find a G.P. for which sum of the first two terms is – 4 and the fifth term is 4 times the third term
Answer:
Given : sum of the first two terms is – 4 and the fifth term is 4 times the third term
Let first term be a and common ratio be r
If r=2, then
If r= - 2, then
Thus, required GP is or
Question:17 If the terms of a G.P. are x, y and z, respectively. Prove that x,y, z are in G.P.
Answer:
Let x,y, z are in G.P.
Let first term=a and common ratio = r
Dividing equation 2 by 1, we have
Dividing equation 3 by 2, we have
Equating values of , we have
Thus, x,y,z are in GP
Question:18 Find the sum to n terms of the sequence, 8, 88, 888, 8888… .
Answer:
8, 88, 888, 8888… is not a GP.
It can be changed in GP by writing terms as
to n terms
Answer:
Here, is a GP.
first term =a=4
common ratio =r
Question:20 Show that the products of the corresponding terms of the sequences form a G.P, and find the common ratio.
Answer:
To prove : is a GP.
Thus, the above sequence is a GP with common ratio of rR.
Answer:
Let first term be a and common ratio be r.
Given : the third term is greater than the first term by 9, and the second term is greater than the by 18.
Dividing equation 2 by 1 , we get
Putting value of r , we get
Thus, four terms of GP are
Question:22 If the terms of a G.P. are a, b and c, respectively. Prove that
Answer:
To prove :
Let A be the first term and R be common ratio.
According to the given information, we have
L.H.S :
=RHS
Thus, LHS = RHS.
Hence proved.
Answer:
Given : First term =a and n th term = b.
Common ratio = r.
To prove :
Then ,
P = product of n terms
Here, is a AP.
Put in equation (2),
Hence proved .
Question:24 Show that the ratio of the sum of first n terms of a G.P. to the sum of terms from term is
Answer:
Let first term =a and common ratio = r.
Since there are n terms from (n+1) to 2n term.
Sum of terms from (n+1) to 2n.
Thus, the required ratio =
Thus, the common ratio of the sum of first n terms of a G.P. to the sum of terms from term is .
Question:25 If a, b, c and d are in G.P. show that
Answer:
If a, b, c and d are in G.P.
To prove :
RHS :
Using equation (1) and (2),
= LHS
Hence proved
Question:26 Insert two numbers between 3 and 81 so that the resulting sequence is G.P.
Answer:
Let A, B be two numbers between 3 and 81 such that series 3, A, B,81 forms a GP.
Let a=first term and common ratio =r.
For ,
The, required numbers are 9,27.
Question:27 Find the value of n so that may be the geometric mean between a and b.
Answer:
M of a and b is
Given :
Squaring both sides ,
Question:28 The sum of two numbers is 6 times their geometric mean, show that numbers are in the ratio
Answer:
Let there be two numbers a and b
geometric mean
According to the given condition,
.............................................................(1)
Also,
.......................................................(2)
From (1) and (2), we get
Putting the value of 'a' in (1),
Thus, the ratio is
Question:29 If A and G be A.M. and G.M., respectively between two positive numbers, prove that the numbers are
Answer:
If A and G be A.M. and G.M., respectively between two positive numbers,
Two numbers be a and b.
...................................................................1
...........................................................................2
We know
Put values from equation 1 and 2,
..................................................................3
From 1 and 3 , we have
Put value of a in equation 1, we get
Thus, numbers are
Answer:
The number of bacteria in a certain culture doubles every hour.It forms GP.
Given : a=30 and r=2.
Thus, bacteria present at the end of the 2nd hour, 4th hour and nth hour are 120,480 and respectively.
Answer:
Given: Bank pays an annual interest rate of 10% compounded annually.
Rs 500 amounts are deposited in the bank.
At the end of the first year, the amount
At the end of the second year, the amount
At the end of the third year, the amount
At the end of 10 years, the amount
Thus, at the end of 10 years, amount
Answer:
Let roots of the quadratic equation be a and b.
According to given condition,
We know that
Thus, the quadratic equation =
Class 11 maths chapter 9 question answer - Exercise: 9.4
Question:1 Find the sum to n terms of each of the series in
Answer:
the series =
n th term =
Question:2 Find the sum to n terms of each of the series in
Answer:
the series =
n th term =
Thus, sum is
Question:3 Find the sum to n terms of each of the series
Answer:
the series
nth term =
Thus, the sum is
Question:4 Find the sum to n terms of each of the series in
Answer:
Series =
.................................
Hence, the sum is
Question:5 Find the sum to n terms of each of the series in
Answer:
series =
n th term =
16th term is
Hence, the sum of the series is 2840.
Question:6 Find the sum to n terms of each of the series
Answer:
series =
=(n th term of 3,6,9,...........) (nth terms of 8,11,14,..........)
n th term =
Hence, sum is
Question:8 Find the sum to n terms of the series in Exercises 8 to 10 whose nth terms is given by
Answer:
nth terms is given by
Question:9 Find the sum to n terms of the series in Exercises 8 to 10 whose nth terms is given by
Answer:
nth terms are given by
This term is a GP with first term =a =2 and common ratio =r =2.
Thus, the sum is
Question:10 Find the sum to n terms of the series in Exercises 8 to 10 whose nth terms is given by
Answer:
nth terms is given by .
Class 11 maths chapter 9 ncert solutions - Miscellaneous Exercise
Question:1 Show that the sum of and terms of an A.P. is equal to twice the term.
Answer:
Let a be first term and d be common difference of AP.
Kth term of a AP is given by,
Hence, the sum of and terms of an A.P. is equal to twice the term.
Question:2 If the sum of three numbers in A.P., is 24 and their product is 440, find the numbers.
Answer:
Let three numbers of AP are a-d, a, a+d.
According to given information ,
When d=3, AP= 5,8,11 also if d=-3 ,AP =11,8,5.
Thus, three numbers are 5,8,11.
Question:3 Let the sum of n, 2n, 3n terms of an A.P. be , respectively, show that
Answer:
Let a be first term and d be common difference of AP.
Subtract equation 1 from 2,
Hence, the result is proved.
Question:4 Find the sum of all numbers between 200 and 400 which are divisible by 7.
Answer:
Numbers divisible by 7 from 200 to 400 are
This sequence is an A.P.
Here , first term =a =203
common difference = 7.
We know ,
The sum of numbers divisible by 7from 200 to 400 is 8729.
Question:5 Find the sum of integers from 1 to 100 that are divisible by 2 or 5.
Answer:
Numbers divisible by 2 from 1 to 100 are
This sequence is an A.P.
Here , first term =a =2
common difference = 2.
We know ,
Numbers divisible by 5 from 1 to 100 are
This sequence is an A.P.
Here , first term =a =5
common difference = 5.
We know ,
Numbers divisible by both 2 and 5 from 1 to 100 are
This sequence is an A.P.
Here , first term =a =10
common difference = 10
We know ,
Thus, the sum of integers from 1 to 100 that are divisible by 2 or 5 is 3050.
Question:6 Find the sum of all two digit numbers which when divided by 4, yields 1 as remainder.
Answer:
Numbers divisible by 4, yield remainder as 1 from 10 to 100 are
This sequence is an A.P.
Here , first term =a =13
common difference = 4.
We know ,
The sum of numbers divisible by 4 yield 1 as remainder from 10 to 100 is 1210.
Question:7 If f is a function satisfying f (x +y) = f(x) f(y) for all x, y N such that f(1) = 3 and
Answer:
Given : f (x +y) = f(x) f(y) for all x, y N such that f(1) = 3
Taking , we have
is forms a GP with first term=3 and common ratio = 3.
Therefore,
Thus, value of n is 4.
Answer:
Let the sum of some terms of G.P. is 315 whose first term and the common ratio are 5 and 2
Therefore,
Thus, the value of n is 6.
Last term of GP=6th term
The last term of GP =160
Answer:
Given: The first term of a G.P. is 1. The sum of the third term and fifth term is 90.
Thus, the common ratio of GP is .
Answer:
Let three terms of GP be
Then, we have
...............................................1
from an AP.
....................................................................2
From equation 1 and 2, we get
If r=2, GP = 8,16,32
If r=0.2, GP= 32,16,8.
Thus, the numbers required are 8,16,32.
Answer:
Let GP be
Number of terms = 2n
According to the given condition,
Let the be GP as
Thus, the common ratio is 4.
Answer:
Given : first term =a=11
Let AP be
Given: The sum of the first four terms of an A.P. is 56.
Also, The sum of the last four terms is 112.
Thus, the number of terms of AP is 11.
Question:13 If then show that a, b, c and d are in G.P.
Answer:
Given :
Taking ,
Taking,
From equation 1 and 2 , we have
Thus, a,b,c,d are in GP.
Question:14 Let S be the sum, P the product and R the sum of reciprocals of n terms in a G.P. Prove that
Answer:
Ler there be a GP
According to given information,
To prove :
LHS :
Hence proved
Question:15 The pth, qth and rth terms of an A.P. are a, b, c, respectively. Show that
Answer:
Given: The pth, qth and rth terms of an A.P. are a, b, c, respectively.
To prove :
Let the first term of AP be 't' and common difference be d
Subtracting equation 2 from 1, we get
Subtracting equation 3 from 2, we get
Equating values of d, from equation 4 and 5, we have
Hence proved.
Question:16 If are in A.P., prove that a, b, c are in A.P.
Answer:
Given: are in A.P.
Thus, a,b,c are in AP.
Question:17 If a, b, c, d are in G.P, prove that are in G.P.
Answer:
Given: a, b, c, d are in G.P.
To prove: are in G.P.
Then we can write,
Let be in GP
LHS:
Hence proved
Thus, are in GP
Answer:
Given: a and b are the roots of
Then,
Also, c, d are roots of
Given: a, b, c, d form a G.P
Let,
From 1 and 2, we get
and
On dividing them,
When , r=2 ,
When , r=-2,
CASE (1) when r=2 and x=1,
i.e. (q + p) : (q – p) = 17:15.
CASE (2) when r=-2 and x=-3,
i.e. (q + p) : (q – p) = 17:15.
Question:19 The ratio of the A.M. and G.M. of two positive numbers a and b, is m : n. Show that
Answer:
Let two numbers be a and b.
According to the given condition,
...................................................................1
We get,
.....................................................2
From 1 and 2, we get
Putting the value of a in equation 1, we have
Question:20 If a, b, c are in A.P.; b, c, d are in G.P. and 1/c , 1/d , 1/e are in A.P. prove that a, c, e are in G.P.
Answer:
Given: a, b, c are in A.P
Also, b, c, d are in G.P.
Also, 1/c, 1/d, 1/e are in A.P
To prove: a, c, e are in G.P. i.e.
From 1, we get
From 2, we get
Putting values of b and d, we get
Thus, a, c, e are in G.P.
Question:21(i) Find the sum of the following series up to n terms:
Answer:
is not a GP.
It can be changed in GP by writing terms as
to n terms
Thus, the sum is
Question:21(ii) Find the sum of the following series up to n terms: .6 +. 66 +. 666+…
Answer:
Sum of 0.6 +0. 66 + 0. 666+….................
It can be written as
to n terms
Question:22 Find the 20th term of the series terms.
Answer:
the series =
Thus, the 20th term of series is 1680
Question:23 Find the sum of the first n terms of the series: 3+ 7 +13 +21 +31 +…
Answer:
The series: 3+ 7 +13 +21 +31 +…..............
n th term =
Question:24 If are the sum of first n natural numbers, their squares and their cubes, respectively, show that
Answer:
To prove :
From given information,
Here ,
Also,
From equation 1 and 2 , we have
Hence proved .
Question:25 Find the sum of the following series up to n terms:
Answer:
n term of series :
Here, are in AP with first term =a=1 , last term = 2n-1, number of terms =n
Question:26 Show that
Answer:
To prove :
the nth term of numerator
nth term of the denominator
Numerator :
Denominator :
From equation 1,2,3,we have
Hence, the above expression is proved.
Answer:
Given : Farmer pays Rs 6000 cash.
Therefore , unpaid amount = 12000-6000=Rs. 6000
According to given condition, interest paid annually is
12% of 6000,12% of 5500,12% of 5000,......................12% of 500.
Thus, total interest to be paid
Here, is a AP with first term =a=500 and common difference =d = 500
We know that
Sum of AP:
Thus, interest to be paid :
Thus, cost of tractor = Rs. 12000+ Rs. 4680 = Rs. 16680
Answer:
Given: Shamshad Ali buys a scooter for Rs 22000.
Therefore , unpaid amount = 22000-4000=Rs. 18000
According to the given condition, interest paid annually is
10% of 18000,10% of 17000,10% of 16000,......................10% of 1000.
Thus, total interest to be paid
Here, is a AP with first term =a=1000 and common difference =d = 1000
We know that
Sum of AP:
Thus, interest to be paid :
Thus, cost of tractor = Rs. 22000+ Rs. 17100 = Rs. 39100
Answer:
The numbers of letters mailed forms a GP :
first term = a=4
common ratio=r=4
number of terms = 8
We know that the sum of GP is
costs to mail one letter are 50 paise.
Cost of mailing 87380 letters
Thus, the amount spent when the 8th set of the letter is mailed is Rs. 43690.
Answer:
Given : A man deposited Rs 10000 in a bank at the rate of 5% simple interest annually.
Interest in fifteen year 10000+ 14 times Rs. 500
Amount in 15 th year
Amount in 20 th year
Answer:
Cost of machine = Rs. 15625
Machine depreciate each year by 20%.
Therefore, its value every year is 80% of the original cost i.e. of the original cost.
Value at the end of 5 years
Thus, the value of the machine at the end of 5 years is Rs. 5120
Answer:
Let x be the number of days in which 150 workers finish the work.
According to the given information, we have
Series is a AP
first term=a=150
common difference= -4
number of terms = x+8
Since x cannot be negative so x=17.
Thus, in 17 days 150 workers finish the work.
Thus, the required number of days = 17+8=25 days.
9.1 Introduction
9.2 Sequences
9.3 Series
9.4 Arithmetic Progression (A.P.)
9.5 Geometric Progression (G.P.)
9.6 Relationship Between A.M. and G.M.
9.7 Sum to n terms of Special Series
If you are interested in Sequence and Series maths chapter 9 class 11 exercises then these are listed below.
chapter-1 | Sets |
chapter-2 | Relations and Functions |
chapter-3 | Trigonometric Functions |
chapter-4 | Principle of Mathematical Induction |
chapter-5 | Complex Numbers and Quadratic equations |
chapter-6 | Linear Inequalities |
chapter-7 | Permutation and Combinations |
chapter-8 | Binomial Theorem |
chapter-9 | Sequences and Series |
chapter-10 | Straight Lines |
chapter-11 | Conic Section |
chapter-12 | Introduction to Three Dimensional Geometry |
chapter-13 | Limits and Derivatives |
chapter-14 | Mathematical Reasoning |
chapter-15 | Statistics |
chapter-16 | Probability |
Easy Language: The class 11 maths ch 9 is written in simple and easy-to-understand language, making it easy for students to comprehend the concepts.
Comprehensive Coverage: The sequences and series ncert solutions provides a comprehensive coverage of all the topics that are enumerated in the syllabus, making it an ideal reference material for students preparing for exams.
Conceptual Clarity: The class 11 maths ch 9 question answer lays a strong foundation for the concepts of sequences and series, providing students with a clear understanding of the fundamentals.
NCERT solutions for class 11 biology |
NCERT solutions for class 11 maths |
NCERT solutions for class 11 chemistry |
NCERT solutions for Class 11 physics |
There are some important formulas from NCERT solutions for class 11 maths chapter 9 sequences and series which you should remember-
There are 32 questions given in a miscellaneous exercise. To get command on this chapter you need to practice more problems, and you can solve miscellaneous exercise for the same. If you are finding difficulties, you can take help of the NCERT solutions for class 11 maths chapter 9 sequences and series which has solutions of miscellaneous exercise too.
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The sequence and series class 11 solutions includes topics such as basic concepts of sequences and series, arithmetic progression (A.P.), geometric progression (G.P.), relationship between A.M. and G.M., and sum to n terms of special series. students can also refer NCERT Syllabus to understand important topics. Also they can practice class 11 maths chapter 9 solutions to get good hold on these concepts.
NCERT solutions are helpful for the students if they stuck while solving the NCERT problems. Also, it will give them new ways to solve the NCERT problems. after solving them, you can have more than one way to solve a problem as well as your confidence level increase which immensely help you to score well in exam.
A progression is a sequence that follows a particular pattern. In Arithmetic Progression (A.P.), any two consecutive terms have a constant difference. To gain a deeper understanding of these concepts, students are advised to refer to the expertly designed NCERT Solutions available at CAreers360. Although the textbook provides sufficient information, utilizing the best study materials can aid in comprehending their practical applications.
Interested students can find a detailed NCERT solutions for class 11 maths by clicking on the link. they can prcatice these solutions to get indepth understanding of the concepts. The sequence and series class 11 ncert solutions are crucial as these are foundation for other chapters such as statistics also questions for this chapter are easy to moderates.
To minimize conceptual errors in class 11th maths chapter 9 Sequence and Series, it is necessary to practice extensively as it comprises several complex topics. Though the initial understanding of fundamental concepts might pose a challenge, achieving high scores is feasible with proper guidance. To excel in class 11 chapter 9 maths , students must tackle a variety of challenging problems. NCERT Solutions provide quick problem-solving techniques and familiarize students with the types of questions that are likely to appear on exams.
Database professionals use software to store and organise data such as financial information, and customer shipping records. Individuals who opt for a career as data administrators ensure that data is available for users and secured from unauthorised sales. DB administrators may work in various types of industries. It may involve computer systems design, service firms, insurance companies, banks and hospitals.
The field of biomedical engineering opens up a universe of expert chances. An Individual in the biomedical engineering career path work in the field of engineering as well as medicine, in order to find out solutions to common problems of the two fields. The biomedical engineering job opportunities are to collaborate with doctors and researchers to develop medical systems, equipment, or devices that can solve clinical problems. Here we will be discussing jobs after biomedical engineering, how to get a job in biomedical engineering, biomedical engineering scope, and salary.
A career as ethical hacker involves various challenges and provides lucrative opportunities in the digital era where every giant business and startup owns its cyberspace on the world wide web. Individuals in the ethical hacker career path try to find the vulnerabilities in the cyber system to get its authority. If he or she succeeds in it then he or she gets its illegal authority. Individuals in the ethical hacker career path then steal information or delete the file that could affect the business, functioning, or services of the organization.
GIS officer work on various GIS software to conduct a study and gather spatial and non-spatial information. GIS experts update the GIS data and maintain it. The databases include aerial or satellite imagery, latitudinal and longitudinal coordinates, and manually digitized images of maps. In a career as GIS expert, one is responsible for creating online and mobile maps.
The invention of the database has given fresh breath to the people involved in the data analytics career path. Analysis refers to splitting up a whole into its individual components for individual analysis. Data analysis is a method through which raw data are processed and transformed into information that would be beneficial for user strategic thinking.
Data are collected and examined to respond to questions, evaluate hypotheses or contradict theories. It is a tool for analyzing, transforming, modeling, and arranging data with useful knowledge, to assist in decision-making and methods, encompassing various strategies, and is used in different fields of business, research, and social science.
Individuals who opt for a career as geothermal engineers are the professionals involved in the processing of geothermal energy. The responsibilities of geothermal engineers may vary depending on the workplace location. Those who work in fields design facilities to process and distribute geothermal energy. They oversee the functioning of machinery used in the field.
If you are intrigued by the programming world and are interested in developing communications networks then a career as database architect may be a good option for you. Data architect roles and responsibilities include building design models for data communication networks. Wide Area Networks (WANs), local area networks (LANs), and intranets are included in the database networks. It is expected that database architects will have in-depth knowledge of a company's business to develop a network to fulfil the requirements of the organisation. Stay tuned as we look at the larger picture and give you more information on what is db architecture, why you should pursue database architecture, what to expect from such a degree and what your job opportunities will be after graduation. Here, we will be discussing how to become a data architect. Students can visit NIT Trichy, IIT Kharagpur, JMI New Delhi.
Individuals who opt for a career as a remote sensing technician possess unique personalities. Remote sensing analysts seem to be rational human beings, they are strong, independent, persistent, sincere, realistic and resourceful. Some of them are analytical as well, which means they are intelligent, introspective and inquisitive.
Remote sensing scientists use remote sensing technology to support scientists in fields such as community planning, flight planning or the management of natural resources. Analysing data collected from aircraft, satellites or ground-based platforms using statistical analysis software, image analysis software or Geographic Information Systems (GIS) is a significant part of their work. Do you want to learn how to become remote sensing technician? There's no need to be concerned; we've devised a simple remote sensing technician career path for you. Scroll through the pages and read.
Budget analysis, in a nutshell, entails thoroughly analyzing the details of a financial budget. The budget analysis aims to better understand and manage revenue. Budget analysts assist in the achievement of financial targets, the preservation of profitability, and the pursuit of long-term growth for a business. Budget analysts generally have a bachelor's degree in accounting, finance, economics, or a closely related field. Knowledge of Financial Management is of prime importance in this career.
The invention of the database has given fresh breath to the people involved in the data analytics career path. Analysis refers to splitting up a whole into its individual components for individual analysis. Data analysis is a method through which raw data are processed and transformed into information that would be beneficial for user strategic thinking.
Data are collected and examined to respond to questions, evaluate hypotheses or contradict theories. It is a tool for analyzing, transforming, modeling, and arranging data with useful knowledge, to assist in decision-making and methods, encompassing various strategies, and is used in different fields of business, research, and social science.
An underwriter is a person who assesses and evaluates the risk of insurance in his or her field like mortgage, loan, health policy, investment, and so on and so forth. The underwriter career path does involve risks as analysing the risks means finding out if there is a way for the insurance underwriter jobs to recover the money from its clients. If the risk turns out to be too much for the company then in the future it is an underwriter who will be held accountable for it. Therefore, one must carry out his or her job with a lot of attention and diligence.
A Product Manager is a professional responsible for product planning and marketing. He or she manages the product throughout the Product Life Cycle, gathering and prioritising the product. A product manager job description includes defining the product vision and working closely with team members of other departments to deliver winning products.
Individuals in the operations manager jobs are responsible for ensuring the efficiency of each department to acquire its optimal goal. They plan the use of resources and distribution of materials. The operations manager's job description includes managing budgets, negotiating contracts, and performing administrative tasks.
Individuals who opt for a career as a stock analyst examine the company's investments makes decisions and keep track of financial securities. The nature of such investments will differ from one business to the next. Individuals in the stock analyst career use data mining to forecast a company's profits and revenues, advise clients on whether to buy or sell, participate in seminars, and discussing financial matters with executives and evaluate annual reports.
A Researcher is a professional who is responsible for collecting data and information by reviewing the literature and conducting experiments and surveys. He or she uses various methodological processes to provide accurate data and information that is utilised by academicians and other industry professionals. Here, we will discuss what is a researcher, the researcher's salary, types of researchers.
Welding Engineer Job Description: A Welding Engineer work involves managing welding projects and supervising welding teams. He or she is responsible for reviewing welding procedures, processes and documentation. A career as Welding Engineer involves conducting failure analyses and causes on welding issues.
A career as Transportation Planner requires technical application of science and technology in engineering, particularly the concepts, equipment and technologies involved in the production of products and services. In fields like land use, infrastructure review, ecological standards and street design, he or she considers issues of health, environment and performance. A Transportation Planner assigns resources for implementing and designing programmes. He or she is responsible for assessing needs, preparing plans and forecasts and compliance with regulations.
Individuals who opt for a career as an environmental engineer are construction professionals who utilise the skills and knowledge of biology, soil science, chemistry and the concept of engineering to design and develop projects that serve as solutions to various environmental problems.
A Safety Manager is a professional responsible for employee’s safety at work. He or she plans, implements and oversees the company’s employee safety. A Safety Manager ensures compliance and adherence to Occupational Health and Safety (OHS) guidelines.
A Conservation Architect is a professional responsible for conserving and restoring buildings or monuments having a historic value. He or she applies techniques to document and stabilise the object’s state without any further damage. A Conservation Architect restores the monuments and heritage buildings to bring them back to their original state.
A Structural Engineer designs buildings, bridges, and other related structures. He or she analyzes the structures and makes sure the structures are strong enough to be used by the people. A career as a Structural Engineer requires working in the construction process. It comes under the civil engineering discipline. A Structure Engineer creates structural models with the help of computer-aided design software.
Highway Engineer Job Description: A Highway Engineer is a civil engineer who specialises in planning and building thousands of miles of roads that support connectivity and allow transportation across the country. He or she ensures that traffic management schemes are effectively planned concerning economic sustainability and successful implementation.
Are you searching for a Field Surveyor Job Description? A Field Surveyor is a professional responsible for conducting field surveys for various places or geographical conditions. He or she collects the required data and information as per the instructions given by senior officials.
Orthotists and Prosthetists are professionals who provide aid to patients with disabilities. They fix them to artificial limbs (prosthetics) and help them to regain stability. There are times when people lose their limbs in an accident. In some other occasions, they are born without a limb or orthopaedic impairment. Orthotists and prosthetists play a crucial role in their lives with fixing them to assistive devices and provide mobility.
A career in pathology in India is filled with several responsibilities as it is a medical branch and affects human lives. The demand for pathologists has been increasing over the past few years as people are getting more aware of different diseases. Not only that, but an increase in population and lifestyle changes have also contributed to the increase in a pathologist’s demand. The pathology careers provide an extremely huge number of opportunities and if you want to be a part of the medical field you can consider being a pathologist. If you want to know more about a career in pathology in India then continue reading this article.
Gynaecology can be defined as the study of the female body. The job outlook for gynaecology is excellent since there is evergreen demand for one because of their responsibility of dealing with not only women’s health but also fertility and pregnancy issues. Although most women prefer to have a women obstetrician gynaecologist as their doctor, men also explore a career as a gynaecologist and there are ample amounts of male doctors in the field who are gynaecologists and aid women during delivery and childbirth.
The audiologist career involves audiology professionals who are responsible to treat hearing loss and proactively preventing the relevant damage. Individuals who opt for a career as an audiologist use various testing strategies with the aim to determine if someone has a normal sensitivity to sounds or not. After the identification of hearing loss, a hearing doctor is required to determine which sections of the hearing are affected, to what extent they are affected, and where the wound causing the hearing loss is found. As soon as the hearing loss is identified, the patients are provided with recommendations for interventions and rehabilitation such as hearing aids, cochlear implants, and appropriate medical referrals. While audiology is a branch of science that studies and researches hearing, balance, and related disorders.
An oncologist is a specialised doctor responsible for providing medical care to patients diagnosed with cancer. He or she uses several therapies to control the cancer and its effect on the human body such as chemotherapy, immunotherapy, radiation therapy and biopsy. An oncologist designs a treatment plan based on a pathology report after diagnosing the type of cancer and where it is spreading inside the body.
Are you searching for an ‘Anatomist job description’? An Anatomist is a research professional who applies the laws of biological science to determine the ability of bodies of various living organisms including animals and humans to regenerate the damaged or destroyed organs. If you want to know what does an anatomist do, then read the entire article, where we will answer all your questions.
For an individual who opts for a career as an actor, the primary responsibility is to completely speak to the character he or she is playing and to persuade the crowd that the character is genuine by connecting with them and bringing them into the story. This applies to significant roles and littler parts, as all roles join to make an effective creation. Here in this article, we will discuss how to become an actor in India, actor exams, actor salary in India, and actor jobs.
Individuals who opt for a career as acrobats create and direct original routines for themselves, in addition to developing interpretations of existing routines. The work of circus acrobats can be seen in a variety of performance settings, including circus, reality shows, sports events like the Olympics, movies and commercials. Individuals who opt for a career as acrobats must be prepared to face rejections and intermittent periods of work. The creativity of acrobats may extend to other aspects of the performance. For example, acrobats in the circus may work with gym trainers, celebrities or collaborate with other professionals to enhance such performance elements as costume and or maybe at the teaching end of the career.
Career as a video game designer is filled with excitement as well as responsibilities. A video game designer is someone who is involved in the process of creating a game from day one. He or she is responsible for fulfilling duties like designing the character of the game, the several levels involved, plot, art and similar other elements. Individuals who opt for a career as a video game designer may also write the codes for the game using different programming languages.
Depending on the video game designer job description and experience they may also have to lead a team and do the early testing of the game in order to suggest changes and find loopholes.
Radio Jockey is an exciting, promising career and a great challenge for music lovers. If you are really interested in a career as radio jockey, then it is very important for an RJ to have an automatic, fun, and friendly personality. If you want to get a job done in this field, a strong command of the language and a good voice are always good things. Apart from this, in order to be a good radio jockey, you will also listen to good radio jockeys so that you can understand their style and later make your own by practicing.
A career as radio jockey has a lot to offer to deserving candidates. If you want to know more about a career as radio jockey, and how to become a radio jockey then continue reading the article.
The word “choreography" actually comes from Greek words that mean “dance writing." Individuals who opt for a career as a choreographer create and direct original dances, in addition to developing interpretations of existing dances. A Choreographer dances and utilises his or her creativity in other aspects of dance performance. For example, he or she may work with the music director to select music or collaborate with other famous choreographers to enhance such performance elements as lighting, costume and set design.
A career as social media manager involves implementing the company’s or brand’s marketing plan across all social media channels. Social media managers help in building or improving a brand’s or a company’s website traffic, build brand awareness, create and implement marketing and brand strategy. Social media managers are key to important social communication as well.
Photography is considered both a science and an art, an artistic means of expression in which the camera replaces the pen. In a career as a photographer, an individual is hired to capture the moments of public and private events, such as press conferences or weddings, or may also work inside a studio, where people go to get their picture clicked. Photography is divided into many streams each generating numerous career opportunities in photography. With the boom in advertising, media, and the fashion industry, photography has emerged as a lucrative and thrilling career option for many Indian youths.
An individual who is pursuing a career as a producer is responsible for managing the business aspects of production. They are involved in each aspect of production from its inception to deception. Famous movie producers review the script, recommend changes and visualise the story.
They are responsible for overseeing the finance involved in the project and distributing the film for broadcasting on various platforms. A career as a producer is quite fulfilling as well as exhaustive in terms of playing different roles in order for a production to be successful. Famous movie producers are responsible for hiring creative and technical personnel on contract basis.
In a career as a copywriter, one has to consult with the client and understand the brief well. A career as a copywriter has a lot to offer to deserving candidates. Several new mediums of advertising are opening therefore making it a lucrative career choice. Students can pursue various copywriter courses such as Journalism, Advertising, Marketing Management. Here, we have discussed how to become a freelance copywriter, copywriter career path, how to become a copywriter in India, and copywriting career outlook.
In a career as a vlogger, one generally works for himself or herself. However, once an individual has gained viewership there are several brands and companies that approach them for paid collaboration. It is one of those fields where an individual can earn well while following his or her passion.
Ever since internet costs got reduced the viewership for these types of content has increased on a large scale. Therefore, a career as a vlogger has a lot to offer. If you want to know more about the Vlogger eligibility, roles and responsibilities then continue reading the article.
For publishing books, newspapers, magazines and digital material, editorial and commercial strategies are set by publishers. Individuals in publishing career paths make choices about the markets their businesses will reach and the type of content that their audience will be served. Individuals in book publisher careers collaborate with editorial staff, designers, authors, and freelance contributors who develop and manage the creation of content.
Careers in journalism are filled with excitement as well as responsibilities. One cannot afford to miss out on the details. As it is the small details that provide insights into a story. Depending on those insights a journalist goes about writing a news article. A journalism career can be stressful at times but if you are someone who is passionate about it then it is the right choice for you. If you want to know more about the media field and journalist career then continue reading this article.
Individuals in the editor career path is an unsung hero of the news industry who polishes the language of the news stories provided by stringers, reporters, copywriters and content writers and also news agencies. Individuals who opt for a career as an editor make it more persuasive, concise and clear for readers. In this article, we will discuss the details of the editor's career path such as how to become an editor in India, editor salary in India and editor skills and qualities.
Individuals who opt for a career as a reporter may often be at work on national holidays and festivities. He or she pitches various story ideas and covers news stories in risky situations. Students can pursue a BMC (Bachelor of Mass Communication), B.M.M. (Bachelor of Mass Media), or MAJMC (MA in Journalism and Mass Communication) to become a reporter. While we sit at home reporters travel to locations to collect information that carries a news value.
Are you searching for a Corporate Executive job description? A Corporate Executive role comes with administrative duties. He or she provides support to the leadership of the organisation. A Corporate Executive fulfils the business purpose and ensures its financial stability. In this article, we are going to discuss how to become corporate executive.
A multimedia specialist is a media professional who creates, audio, videos, graphic image files, computer animations for multimedia applications. He or she is responsible for planning, producing, and maintaining websites and applications.
Welding Engineer Job Description: A Welding Engineer work involves managing welding projects and supervising welding teams. He or she is responsible for reviewing welding procedures, processes and documentation. A career as Welding Engineer involves conducting failure analyses and causes on welding issues.
A quality controller plays a crucial role in an organisation. He or she is responsible for performing quality checks on manufactured products. He or she identifies the defects in a product and rejects the product.
A quality controller records detailed information about products with defects and sends it to the supervisor or plant manager to take necessary actions to improve the production process.
A Product Manager is a professional responsible for product planning and marketing. He or she manages the product throughout the Product Life Cycle, gathering and prioritising the product. A product manager job description includes defining the product vision and working closely with team members of other departments to deliver winning products.
A QA Lead is in charge of the QA Team. The role of QA Lead comes with the responsibility of assessing services and products in order to determine that he or she meets the quality standards. He or she develops, implements and manages test plans.
A Structural Engineer designs buildings, bridges, and other related structures. He or she analyzes the structures and makes sure the structures are strong enough to be used by the people. A career as a Structural Engineer requires working in the construction process. It comes under the civil engineering discipline. A Structure Engineer creates structural models with the help of computer-aided design software.
The Process Development Engineers design, implement, manufacture, mine, and other production systems using technical knowledge and expertise in the industry. They use computer modeling software to test technologies and machinery. An individual who is opting career as Process Development Engineer is responsible for developing cost-effective and efficient processes. They also monitor the production process and ensure it functions smoothly and efficiently.
An AWS Solution Architect is someone who specializes in developing and implementing cloud computing systems. He or she has a good understanding of the various aspects of cloud computing and can confidently deploy and manage their systems. He or she troubleshoots the issues and evaluates the risk from the third party.
An Azure Administrator is a professional responsible for implementing, monitoring, and maintaining Azure Solutions. He or she manages cloud infrastructure service instances and various cloud servers as well as sets up public and private cloud systems.
Careers in computer programming primarily refer to the systematic act of writing code and moreover include wider computer science areas. The word 'programmer' or 'coder' has entered into practice with the growing number of newly self-taught tech enthusiasts. Computer programming careers involve the use of designs created by software developers and engineers and transforming them into commands that can be implemented by computers. These commands result in regular usage of social media sites, word-processing applications and browsers.
A Product Manager is a professional responsible for product planning and marketing. He or she manages the product throughout the Product Life Cycle, gathering and prioritising the product. A product manager job description includes defining the product vision and working closely with team members of other departments to deliver winning products.
Individuals in the information security manager career path involves in overseeing and controlling all aspects of computer security. The IT security manager job description includes planning and carrying out security measures to protect the business data and information from corruption, theft, unauthorised access, and deliberate attack
An Automation Test Engineer job involves executing automated test scripts. He or she identifies the project’s problems and troubleshoots them. The role involves documenting the defect using management tools. He or she works with the application team in order to resolve any issues arising during the testing process.
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