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Sequences and patterns are significant parts of fields such as finance, science, and architecture, among others. Consider an example where growth of investments over time is calculated and analysis of population increase is done. In such scenarios, the concept of geometric progression is used. Along with this, concepts like arithmetic mean and geometric mean are widely used in statistics, economics, and data analysis to find central tendencies.
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JEE Main Scholarship Test Kit (Class 11): Narayana | Physics Wallah | Aakash | Unacademy
Suggested: JEE Main: high scoring chapters | Past 10 year's papers
Class 11 Maths Chapter 8 Miscellaneous Exercise of NCERT provides a mix of problems drawn from all the topics covered in the chapter. These include arithmetic and geometric progressions, sums of series, and special series. If you are looking for NCERT Solutions, you can click on the given link to get NCERT Solutions for Classes 6 to 12.
Question 1: If f is a function satisfying f (x +y) = f(x) f(y) for all x, y
Answer:
Given : f (x +y) = f(x) f(y) for all x, y
Taking
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
Question 3: The first term of a G.P. is 1. The sum of the third term and fifth term is 90. Find the common ratio of G.P.
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
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.
Question 6: If
Answer:
Given :
Taking ,
Taking,
From equation 1 and 2 , we have
Thus, a,b,c,d are in GP.
Question 7: 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 8: If a, b, c, d are in G.P, prove that
Answer:
Given: a, b, c, d are in G.P.
To prove:
Then we can write,
Let
LHS:
Hence proved
Thus,
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
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 10: 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,
We get,
From 1 and 2, we get
Putting the value of a in equation 1, we have
Question 11:(i) Find the sum of the following series up to n terms:
Answer:
It can be changed in GP by writing terms as
Thus, the sum is
Question 11:(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
Question 12: Find the 20th term of the series
Answer:
the series =
Thus, the 20th term of series is 1680
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,
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,
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.
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.
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
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.
Also Read
Arithmetic Progression (A.P.)
In an arithmetic progression, the difference between any two consecutive terms is constant. It is commonly used in daily life situations like calculating savings, salaries, or planning installments.
Geometric Progression (G.P.)
A geometric progression is a sequence where each term is found by multiplying the previous term by a fixed number. It is useful in problems involving growth, decay, and compound interest.
Relationship Between A.M. and G.M.
The Arithmetic Mean (A.M.) and Geometric Mean (G.M.) are measures of central tendency. This topic explores how they relate to each other and the inequality that connects them.
Sum to n terms of Special Series
This includes formulas and techniques to find the sum of series like natural numbers, squares, and cubes. These are important in simplifying large summations and solving advanced problems.
Also Read
Students can also access the NCERT solutions for other subjects and make their learning feasible.
NCERT Solutions for Class 11 Maths |
NCERT Solutions for Class 11 Physics |
NCERT Solutions for Class 11 Chemistry |
NCERT Solutions for Class 11 Biology |
Use the links provided in the table below to get your hands on the NCERT exemplar solutions available for all the subjects.
NCERT Exemplar Solutions for Class 11 Maths |
NCERT Exemplar Solutions for Class 11 Physics |
NCERT Exemplar Solutions for Class 11 Chemistry |
NCERT Exemplar Solutions for Class 11 Biology |
Sequence means an arrangement of numbers in a definite order according to some rule.
The finite sequence is s sequence containing a finite number of terms.
The infinite sequence is s sequence containing an infinite number of terms.
If the terms of a sequence are expressed as the sum of terms then it is called a series.
Sum of A.P. = n(a+l)/2
Arithmetic mean (A.M.) = (5+9)/2 = 7
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