NCERT Solutions for Exercise 9.3 Class 11 Maths Chapter 9 - Sequences and Series

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NCERT Solutions for Exercise 9.3 Class 11 Maths Chapter 9 - Sequences and Series

Edited By Vishal kumar | Updated on Nov 08, 2023 10:28 AM IST

NCERT Solutions for Class 11 Maths Chapter 9: Sequences and Series Exercise 9.3- Download Free PDF

NCERT Solutions for Class 11 Maths Chapter 9: Sequences and Series Exercise 9.3-In the previous exercise, you have already learned about the arithmetic progression, nth term and sum of the arithmetic progression, arithmetic mean, etc. In NCERT syllabus Class 11 Maths chapter 9 exercise 9.3, you will solve questions on geometric progression, nth term of the geometric progression, the sum of the geometric progression, geometric mean, relationship between the arithmetic mean and geometric mean, etc. You must have learned about the geometric progression, the nth term of the geometric progression, and the sum of the terms of the geometric progression in the previous classes. As you are familiar with these topics, it won't take much effort to understand these topics.

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NCERT Solutions for Class 11 Maths Chapter 9: Sequences and Series Exercise 9.3- Download Free PDF
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Question:1 Find the and terms of the G.P.
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You will learn some new topics like arithmetic mean, geometric mean and the relationship between the arithmetic mean and geometric mean in the exercise 9.3 Class 11 Maths. Arithmetic mean and geometric mean are very similar to arithmetic progression and geometric progression respectively, so you will understand them very easily. Check NCERT Solutions if you are looking for NCERT solutions for Class 6 to Class 12 at one place. All the solutions are expertly crafted by subject experts at Careers360 in a user-friendly and detailed manner. Furthermore, PDF versions of these solutions are available for free, providing students with convenient access.

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**As per the CBSE Syllabus for 2023-24, this chapter has been renumbered as Chapter 8.

Download the PDF of NCERT Solutions for Class 11 Maths Chapter 9: Sequences and Series Exercise 9.3

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Access Sequences And Series Class 11 Chapter 9 Exercise 9.3

Question:1 Find the $20 ^{th}$ and $n ^{th}$ terms of the G.P. $\frac{5}{2},\frac{5}{4},\frac{5}{8},....$

Answer:

G.P :

$\frac{5}{2},\frac{5}{4},\frac{5}{8},....$

first term = a

$a=\frac{5}{2}$

common ratio =r

$r=\frac{\frac{5}{4}}{\frac{5}{2}}=\frac{1}{2}$

$a_n=a.r^{n-1}$

$a_2_0=\frac{5}{2}.(\frac{1}{2})^{20-1}$

$a_2_0=\frac{5}{2}.(\frac{1}{2^{19}})$

$a_2_0=\frac{5}{2^{20}}$

$a_n=a.r^{n-1}$

$a_n=\frac{5}{2}.\left ( \frac{1}{2} \right )^{n-1}$

$a_n=\frac{5}{2}. \frac{1}{2^{n-1}}$

$a_n=\frac{5}{2^n}$ the nth term of G.P

Question:2 Find the $12 ^{th}$ term of a G.P. whose $8 ^{th}$ term is 192 and the common ratio is 2.

Answer:

First term = a

common ratio =r=2

$8 ^{th}$ term is 192

$a_n=a.r^{n-1}$

$a_8=a.(2)^{8-1}$

$192=a.(2)^{7}$

$a=\frac{2^6.3}{2^7}$

$a=\frac{3}{2}$

$a_n=a.r^{n-1}$

$a_1_2=\frac{3}{2}. ( 2 )^{12-1}$

$a_1_2=\frac{3}{2}. ( 2 )^{11}$

$a_1_2= 3. ( 2 )^{10}$

$a_1_2= 3072$ is the $12 ^{th}$ term of a G.P.

Question:3 The $5 ^{th} , 8 ^{th} \: \:and \: \: 11 ^{th}$ terms of a G.P. are p, q and s, respectively. Show that $q ^2 = ps$

Answer:

To prove : $q ^2 = ps$

Let first term=a and common ratio = r

$a_5=a.r^4=p..................(1)$

$a_8=a.r^7=q..................(2)$

$a_1_1=a.r^1^0=s..................(3)$

Dividing equation 2 by 1, we have

$\frac{a.r^7}{a.r^4}=\frac{q}{p}$

$\Rightarrow r^3=\frac{q}{p}$

Dividing equation 3 by 2, we have

$\frac{a.r^1^0}{a.r^7}=\frac{s}{q}$

$\Rightarrow r^3=\frac{s}{q}$

Equating values of $r^3$ , we have

$\frac{q}{p}=\frac{s}{q}$

$\Rightarrow q^2=ps$

Hence proved

Question:4 The $4^{th}$ term of a G.P. is square of its second term, and the first term is -3. Determine its $7^{th}$ term.

Answer:

First term =a= -3

$4^{th}$ term of a G.P. is square of its second term

$\Rightarrow a_4=(a_2)^2$

$\Rightarrow a.r^{4-1}=(a.r^{2-1})^2$

$\Rightarrow a.r^{3}=a^2.r^{2}$

$\Rightarrow r=a=-3$

$a_7=a.r^{7-1}$

$\Rightarrow a_7=(-3).(-3)^{6}$

$\Rightarrow a_7=(-3)^{7}=-2187$

Thus, seventh term is -2187.

Question:5(a) Which term of the following sequences: $2,2\sqrt 2 , 4 .,....is \: \: 128 ?$

Answer:

Given : $GP = 2,2\sqrt 2 , 4 .,............$

$a=2\, \, \, \, \, and \, \, \, \, \, r=\frac{2\sqrt{2}}{2}=\sqrt{2}$

n th term is given as 128.

$a_n=a.r^{n-1}$

$\Rightarrow 128=2.(\sqrt{2})^{n-1}$

$\Rightarrow 64=(\sqrt{2})^{n-1}$

$\Rightarrow 2^6=(\sqrt{2})^{n-1}$

$\Rightarrow \sqrt{2}^1^2=(\sqrt{2})^{n-1}$

$\Rightarrow n-1=12$

$\Rightarrow n=12+1=13$

The, 13 th term is 128.

Question:5(b) Which term of the following sequences: $\sqrt 3 ,3 , 3 \sqrt 3 ,...is \: \: 729 ?$

Answer:

Given : $GP=\sqrt 3 ,3 , 3 \sqrt 3 ,........$

$a=\sqrt{3}\, \, \, \, \, and \, \, \, \, \, r=\frac{3}{\sqrt{3}}=\sqrt{3}$

n th term is given as 729.

$a_n=a.r^{n-1}$

$\Rightarrow 729=\sqrt{3}.(\sqrt{3})^{n-1}$

$\Rightarrow 729=(\sqrt{3})^{n}$

$\Rightarrow( \sqrt{3})^1^2=(\sqrt{3})^{n}$

$\Rightarrow n=12$

The, 12 th term is 729.

Question:5(c) Which term of the following sequences: $\frac{1}{3} , \frac{1}{9} , \frac{1}{27} ,....is \: \: \frac{1}{19683}?$

Answer:

Given : $GP=\frac{1}{3} , \frac{1}{9} , \frac{1}{27} ,............$

$a=\frac{1}{3}\, \, \, \, \, and \, \, \, \, \, r=\frac{\frac{1}{9}}{\frac{1}{3}}=\frac{1}{3}$

n th term is given as $\frac{1}{19683}$

$a_n=a.r^{n-1}$

$\Rightarrow \frac{1}{19683}=\frac{1}{3}.(\frac{1}{3})^{n-1}$

$\Rightarrow \frac{1}{19683}=\frac{1}{3^n}$

$\Rightarrow \frac{1}{3^9}=\frac{1}{3^n}$

$\Rightarrow n=9$

Thus, n=9.

Question:6 For what values of x, the numbers $-\frac{2}{7} ,x, -\frac{7}{2}$ are in G.P.?

Answer:

$GP=-\frac{2}{7} ,x, -\frac{7}{2}$

Common ratio=r.

$r=\frac{x}{\frac{-2}{7}}=\frac{\frac{-7}{2}}{x}$

$\Rightarrow x^2=1$

$\Rightarrow x=\pm 1$

Thus, for $x=\pm 1$ ,given numbers will be in GP.

Question:7 Find the sum to indicated number of terms in each of the geometric progressions in 0.15, 0.015, 0.0015, ... 20 terms.

Answer:

geometric progressions is 0.15, 0.015, 0.0015, ... .....

a=0.15 , r = 0.1 , n=20

$S_n=\frac{a(1-r^n)}{1-r}$

$S_2_0=\frac{0.15(1-(0.1)^{20})}{1-0.1}$

$S_2_0=\frac{0.15(1-(0.1)^{20})}{0.9}$

$S_2_0=\frac{0.15}{0.9}(1-(0.1)^{20})$

$S_2_0=\frac{15}{90}(1-(0.1)^{20})$

$S_2_0=\frac{1}{6}(1-0.1^{20})$

Question:8 Find the sum to indicated number of terms in each of the geometric progressions in $\sqrt 7 , \sqrt {21} , 3 \sqrt 7 ,.... n \: \: terms$

Answer:

$GP=\sqrt 7 , \sqrt {21} , 3 \sqrt 7 ,...............$

$a=\sqrt{7}\, \, \, \, and\, \, \, \, \, r=\frac{\sqrt{21}}{\sqrt{7}}=\sqrt{3}$

$S_n=\frac{a(1-r^n)}{1-r}$

$S_n=\frac{\sqrt{7}(1-\sqrt{3}^n)}{1-\sqrt{3}}$

$S_n=\frac{\sqrt{7}(1-\sqrt{3}^n)}{1-\sqrt{3}}\times \frac{1+\sqrt{3}}{1+\sqrt{3}}$

$S_n=\frac{\sqrt{7}(1-\sqrt{3}^n)}{1-3} (1+\sqrt{3})$

$S_n=\frac{\sqrt{7}(1-\sqrt{3}^n)}{-2} (1+\sqrt{3})$

$S_n=\frac{\sqrt{7}(1+\sqrt{3})}{2} (\sqrt{3}^n-1)$

Question:9 Find the sum to indicated number of terms in each of the geometric progressions in $-a , a^2 , - a ^3 , ... n terms ( if a \neq -1)$

Answer:

The sum to the indicated number of terms in each of the geometric progressions is:

$GP=1,-a , a^2 , - a ^3 , .............$

$a=1\, \, \, and\, \, \, \, r=-a$

$S_n=\frac{a(1-r^n)}{1-r}$

$S_n=\frac{1(1-(-a)^n)}{1-(-a)}$

$S_n=\frac{1(1-(-a)^n)}{1+a}$

$S_n=\frac{1-(-a)^n}{1+a}$

Question:10 Find the sum to indicated number of terms in each of the geometric progressions in $x ^3 , x^5 , x^7 ... n \: \: terms (if \: \: x \neq \pm 1)$

Answer:

$a=x^3\, \, \, and\, \, r=\frac{x^5}{x^3}=x^2$

$S_n=\frac{a(1-r^n)}{1-r}$

$S_n=\frac{x^3(1-(x^2)^n)}{1-x^2}$

$S_n=\frac{x^3(1-x^2^n)}{1-x^2}$

Question:11 Evaluate $\sum_{k = 1}^{11} ( 2+ 3 ^k )$

Answer:

Given : $\sum_{k = 1}^{11} ( 2+ 3 ^k )=\sum _{k=1}^{11}2 +\sum _{k=1}^{11} 3^k$

$=22 +\sum _{k=1}^{11} 3^k...............(1)$

$\sum _{k=1}^{11} 3^k=3^1+3^2+3^3+....................3^1^1$

These terms form GP with a=3 and r=3.

$S_n=\frac{a(1-r^n)}{1-r}$

$S_n=\frac{3(1-3^1^1)}{1-3}$

$S_n=\frac{3(1-3^1^1)}{-2}$

$S_n=\frac{3(3^1^1-1)}{2}$ $=\sum _{k=1}^{11} 3^k$

$\sum_{k = 1}^{11} ( 2+ 3 ^k )$ $=22+\frac{3(3^1^1-1)}{2}$

Question:12 The sum of first three terms of a G.P. is $\frac{39}{10}$ 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 $\frac{39}{10}$ and their product is 1.

Let three terms be $\frac{a}{r},a,ar$ .

$S_n=\frac{a(1-r^n)}{1-r}$

$\frac{a}{r}+a+ar=\frac{39}{10}.........1$

Product of 3 terms is 1.

$\frac{a}{r}\times a\times ar=1$

$\Rightarrow a^3=1$

$\Rightarrow a=1$

Put value of a in equation 1,

$\frac{1}{r}+1+r=\frac{39}{10}$

The three terms of AP are $\frac{5}{2},1,\frac{2}{5}$ .

Question:13 How many terms of G.P. $3 , 3 ^ 2 , 3 ^ 3$ , … are needed to give the sum 120?

Answer:

G.P.= $3 , 3 ^ 2 , 3 ^ 3$ , …............

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.

Question:14 The sum of first three terms of a G.P. is 16 and the sum of the next three terms is 128. Determine the first term, the common ratio and the sum to n terms of the G.P.

Answer:

Let GP be $a,ar,ar^2,ar^3,ar^4,ar^5,ar^6................................$

Given : The sum of first three terms of a G.P. is 16

$a+ar+ar^2=16$

$\Rightarrow a(1+r+r^2)=16...............................(1)$

Given : the sum of the next three terms is128.

$ar^3+ar^4+ar^5=128$

$\Rightarrow ar^3(1+r+r^2)=128...............................(2)$

Dividing equation (2) by (1), we have

$\Rightarrow \frac{ar^3(1+r+r^2)}{a(1+r+r^2)}=\frac{128}{16}$

$\Rightarrow r^3=8$

$\Rightarrow r^3=2^3$

$\Rightarrow r=2$

Putting value of r =2 in equation 1,we have

$\Rightarrow a(1+2+2^2)=16$

$\Rightarrow a(7)=16$

$\Rightarrow a=\frac{16}{7}$

$S_n=\frac{a(1-r^n)}{1-r}$

$S_n=\frac{\frac{16}{7}(1-2^n)}{1-2}$

$S_n=\frac{16}{7}(2^n-1)$

Question:15 Given a G.P. with a = 729 and $7 ^{th}$ term 64, determine $s_7$

Answer:

Given a G.P. with a = 729 and $7 ^{th}$ term 64.

$a_n=a.r^{n-1}$

$\Rightarrow 64=729.r^{7-1}$

$\Rightarrow r^6=\frac{64}{729}$

$\Rightarrow r^6=\left ( \frac{2}{3} \right )^6$

$\Rightarrow r=\frac{2}{3}$

$S_n=\frac{a(1-r^n)}{1-r}$

$S_7=\frac{729(1-\left ( \frac{2}{3} \right )^7)}{1-\frac{2}{3}}$

$S_7=\frac{729(1-\left ( \frac{2}{3} \right )^7)}{\frac{1}{3}}$

$S_7=3\times 729 \left ( \frac{3^7-2^7}{3^7} \right )$

$S_7= \left ( 3^7-2^7 \right )$

$S_7= 2187-128$

$S_7= 2059$ (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

$a_5=4.a_3$

$\Rightarrow a.r^{5-1}=4.a.r^{3-1}$

$\Rightarrow a.r^{4}=4.a.r^{2}$

$\Rightarrow r^{2}=4$

$\Rightarrow r=\pm 2$

If r=2, then

$S_n=\frac{a(1-r^n)}{1-r}$

$\Rightarrow \frac{a(1-2^2)}{1-2}=-4$

$\Rightarrow \frac{a(1-4)}{-1}=-4$

$\Rightarrow a(-3)=4$

$\Rightarrow a=\frac{-4}{3}$

If r= - 2, then

$S_n=\frac{a(1-r^n)}{1-r}$

$\Rightarrow \frac{a(1-(-2)^2)}{1-(-2)}=-4$

$\Rightarrow \frac{a(1-4)}{3}=-4$

$\Rightarrow a(-3)=-12$

$\Rightarrow a=\frac{-12}{-3}=4$

Thus, required GP is $\frac{-4}{3},\frac{-8}{3},\frac{-16}{3},.........$ or $4,-8,-16,-32,..........$

Question:17 If the $4 ^{th} , 10 ^{th} , 16 ^ {th}$ 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

$a_4=a.r^3=x..................(1)$

$a_1_0=a.r^9=y..................(2)$

$a_1_6=a.r^1^5=z..................(3)$

Dividing equation 2 by 1, we have

$\frac{a.r^9}{a.r^3}=\frac{y}{x}$

$\Rightarrow r^4=\frac{y}{x}$

Dividing equation 3 by 2, we have

$\frac{a.r^1^5}{a.r^9}=\frac{z}{y}$

$\Rightarrow r^4=\frac{z}{y}$

Equating values of $r^4$ , we have

$\frac{y}{x}=\frac{z}{y}$

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

$S_n=8+88+888+8888+.............$ to n terms

$S_n=\frac{8}{9}[9+99+999+9999+................]$

$S_n=\frac{8}{9}[(10-1)+(10^2-1)+(10^3-1)+(10^4-1)+................]$

$S_n=\frac{8}{9}[(10+10^2+10^3+........)-(1+1+1.....................)]$

$S_n=\frac{8}{9}[\frac{10(10^n-1)}{10-1}-(n)]$

$S_n=\frac{8}{9}[\frac{10(10^n-1)}{9}-(n)]$

$S_n=\frac{80}{81}(10^n-1)-\frac{8n}{9}$

Question:19 Find the sum of the products of the corresponding terms of the sequences 2, 4, 8, 16, 32 and 128, 32, 8, 2,1/2

Answer:

$Required \, \, sum=2\times 128+4\times 32+8\times 8+16\times 2+32\times \frac{1}{2}$

$Required \, \, sum=64\left [ 4+2+1+\frac{1}{2}+\frac{1}{2^2} \right ]$

Here, $4,2,1,\frac{1}{2},\frac{1}{2^2}$ is a GP.

first term =a=4

common ratio =r

$r=\frac{1}{2}$

$S_n=\frac{a(1-r^n)}{1-r}$

$S_5=\frac{4(1-\left ( \frac{1}{2} \right )^5)}{1-\frac{1}{2}}$

$S_5=\frac{4(1-\left ( \frac{1}{2} \right )^5)}{\frac{1}{2}}$

$S_5=8(1-\left ( \frac{1}{32} \right ))$

$S_5=8(\frac{31}{32})$

$S_5=\frac{31}{4}$

$Required \, \, sum=64\left [ \frac{31}{4} \right ]$

$Required \, \, sum=16\times 31=496$

Question:20 Show that the products of the corresponding terms of the sequences $a,ar, ar^2 , ...ar^{n-1} \: \: and\: \: A ,AR, AR^2 ....AR^{n-1}$ form a G.P, and find the common ratio.

Answer:

To prove : $aA,arAR,ar^2AR^2,...................$ is a GP.

$\frac{second \, \, term}{first\, \, term}=\frac{arAR}{aA}=rR$

$\frac{third \, \, term}{second\, \, term}=\frac{ar^2AR^2}{arAR}=rR$

Thus, the above sequence is a GP with common ratio of rR.

Question:21 Find four numbers forming a geometric progression in which the third term is greater than the first term by 9, and the second term is greater than the $4 ^{th}$ by 18.

Answer:

Let first term be a and common ratio be r.

$a_1=a,a_2=ar,a_3=ar^2,a_4=ar^3$

Given : the third term is greater than the first term by 9, and the second term is greater than the $4 ^{th}$ by 18.

$a_3=a_1+9$

$\Rightarrow ar^2=a+9$

$\Rightarrow a(r^2-1)=9.................1$

$a_2=a_4+18$

$\Rightarrow ar=ar^3+18$

$\Rightarrow ar(1-r^2)=18......................2$

Dividing equation 2 by 1 , we get

$\frac{ ar(1-r^2)}{ -a(1-r^2)}=\frac{18}{9}$

$\Rightarrow r=-2$

Putting value of r , we get

$4a=a+9$

$\Rightarrow 4a-a=9$

$\Rightarrow 3a=9$

$\Rightarrow a=3$

Thus, four terms of GP are $3,-6,12,-24.$

Question:22 If the $p^{th} , q ^{th} , r ^{th}$ terms of a G.P. are a, b and c, respectively. Prove that $a ^{ q-r } b ^{r- p } C ^{p-q} = 1$

Answer:

To prove : $a ^{ q-r } b ^{r- p } C ^{p-q} = 1$

Let A be the first term and R be common ratio.

According to the given information, we have

$a_p=A.R^{p-1}=a$

$a_q=A.R^{q-1}=b$

$a_r=A.R^{r-1}=c$

L.H.S : $a ^{ q-r } b ^{r- p } C ^{p-q}$

$=A^{q-r}.R^{(q-r)(p-1)}.A^{r-p}.R^{(r-p)(q-1)}.A^{p-q}.R^{(p-q)(r-1)}$

$=A^{q-r+r-p+p-q}.R^{(qp-rp-q+r)+(rq-pq+p-r)+(pr-p-qr+q)}$

$=A^0.R^0=1$ =RHS

Thus, LHS = RHS.

Hence proved.

Question:23 If the first and the nth term of a G.P. are a and b, respectively, and if P is the product of n terms, prove that $P^2 = ( ab)^n$ .

Answer:

Given : First term =a and n th term = b.

Common ratio = r.

To prove : $P^2 = ( ab)^n$

Then , $GP = a,ar,ar^2,ar^3,ar^4,..........................$

$a_n=a.r^{n-1}=b..................................1$

P = product of n terms

$P=(a).(ar).(ar^2).(ar^3)..............(ar^{n-1})$

$P=(a.a.a...............a)((1).(r).(r^2).(r^3)..............(r^{n-1}))$

$P=(a^n)(r^{1+2+.........(n-1)})........................................2$

Here, $1+2+.........(n-1)$ is a AP.

$\therefore\, \, \, sum= \frac{n}{2}\left [2a+(n-1)d \right ]$

$= \frac{n-1}{2}\left [2(1)+(n-1-1)1 \right ]$

$= \frac{n-1}{2}\left [2+n-2 \right ]$

$= \frac{n-1}{2}\left [n \right ]$

$= \frac{n(n-1)}{2}$

Put in equation (2),

$P=(a^n)(r^{\frac{n(n-1)}{2}})$

$P^2=(a^2^n)(r^{n(n-1)})$

$P^2=(a. a.r^{(n-1)})^n$

$P^2=(a.b)^n$

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 $( n+1)^{th} \: \: to\: \: (2n)^{th}$ term is $\frac{1}{r^n}$

Answer:

Let first term =a and common ratio = r.

$sum \, \, of\, \, n\, \, terms=\frac{a(1-r^n)}{1-r}$

Since there are n terms from (n+1) to 2n term.

Sum of terms from (n+1) to 2n.

$S_n=\frac{a_(_n+_1_)(1-r^n)}{1-r}$

$a_(_n+_1)=a.r^{n+1-1}=ar^n$

Thus, the required ratio = $\frac{a(1-r^n)}{1-r}\times \frac{1-r}{ar^n(1-r^n)}$

$=\frac{1}{r^n}$

Thus, the common ratio of the sum of first n terms of a G.P. to the sum of terms from $( n+1)^{th} \: \: to\: \: (2n)^{th}$ term is $\frac{1}{r^n}$ .

Question:25 If a, b, c and d are in G.P. show that $(a^2 + b^2 + c^2) (b^2 + c^2 + d^2) = (ab + bc + cd)^2 .$

Answer:

If a, b, c and d are in G.P.

$bc=ad....................(1)$

$b^2=ac....................(2)$

$c^2=bd....................(3)$

To prove : $(a^2 + b^2 + c^2) (b^2 + c^2 + d^2) = (ab + bc + cd)^2 .$

RHS : $(ab + bc + cd)^2 .$

$=(ab + ad + cd)^2 .$

$=(ab + d (a+ c))^2 .$

$=a^2b^2 + d^2 (a+ c)^2 + 2(ab)(d(a+c))$

$=a^2b^2 + d^2 (a^2+ c^2+2ac) + 2a^2bd+2bcd$

Using equation (1) and (2),

$=a^2b^2 + 2a^2c^2+ 2b^2c^2+d^2a^2+2d^2b^2+d^2c^2$

$=a^2b^2 + a^2c^2+ a^2c^2+b^2c^2+b^2c^2+d^2a^2+d^2b^2+d^2b^2+d^2c^2$

$=a^2b^2 + a^2c^2+ a^2d^2+b^2.b^2+b^2c^2+b^2d^2+c^2b^2+c^2.c^2+d^2c^2$

$=a^2(b^2 + c^2+ d^2)+b^2(b^2+c^2+d^2)+c^2(b^2+c^2+d^2)$

$=(b^2 + c^2+ d^2)(a^2+b^2+c^2)$ = 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.

$\therefore a_4=a.r^{4-1}$

$81=3.r^{3}$

$27=r^{3}$

$r=3$

For $r=3$ ,

$A=ar=(3)(3)=9$

$B=ar^2=(3)(3)^2=27$

The, required numbers are 9,27.

Question:27 Find the value of n so that $\frac{a^{n+1}+ b ^{n+1}}{a^n+b^n}$ may be the geometric mean between a and b.

Answer:

M of a and b is $\sqrt{ab}.$

Given :

$\frac{a^{n+1}+ b ^{n+1}}{a^n+b^n}=\sqrt{ab}$

Squaring both sides ,

$\left ( \frac{a^{n+1}+ b ^{n+1}}{a^n+b^n} \right )^2=ab$

$\left (a^{n+1}+ b ^{n+1})^2=({a^n+b^n} \right )^2ab$

$\Rightarrow \left (a^{2n+2}+ b ^{2n+2}+2.a^{n+1}.b^{n+1})=({a^2^n+b^2^n+2.a^n.b^n} \right )ab$

$\Rightarrow \left (a^{2n+2}+ b ^{2n+2}+2.a^{n+1}.b^{n+1})=({a^{2n+1}.b+a.b^{2n+1}+2.a^{n+1}.b^{n+1}} \right )$

$\Rightarrow \left (a^{2n+2}+ b ^{2n+2})=({a^{2n+1}.b+a.b^{2n+1}} \right )$

$\Rightarrow a^{2n+2}-{a^{2n+1}.b=a.b^{2n+1}} )- b ^{2n+2}$

$\Rightarrow a^{2n+1}(a-b)=b^{2n+1}( a-b)$

$\Rightarrow a^{2n+1}=b^{2n+1}$

$\Rightarrow \left ( \frac{a}{b} \right )^{2n+1}=1$

$\Rightarrow \left ( \frac{a}{b} \right )^{2n+1}=1=\left ( \frac{a}{b} \right )^0$

$\Rightarrow 2n+1=0$

$\Rightarrow 2n=-1$

$\Rightarrow n=\frac{-1}{2}$

Question:28 The sum of two numbers is 6 times their geometric mean, show that numbers are in the ratio $( 3+ 2 \sqrt 2 ) : ( 3 - 2 \sqrt 2 )$

Answer:

Let there be two numbers a and b

geometric mean $=\sqrt{ab}$

According to the given condition,

$a+b=6\sqrt{ab}$

$(a+b)^2=36(ab)$ .............................................................(1)

Also, $(a-b)^2=(a+b)^2-4ab=36ab-4ab=32ab$

$(a-b)=\sqrt{32}\sqrt{ab}$

$(a-b)=4\sqrt{2}\sqrt{ab}$ .......................................................(2)

From (1) and (2), we get

$2a=(6+4\sqrt{2})\sqrt{ab}$

$a=(3+2\sqrt{2})\sqrt{ab}$

Putting the value of 'a' in (1),

$b=6\sqrt{ab}-(3+2\sqrt{2})\sqrt{ab}$

$b=(3-2\sqrt{2})\sqrt{ab}$

$\frac{a}{b}=\frac{(3+2\sqrt{2})\sqrt{ab}}{(3-2\sqrt{2})\sqrt{ab}}$

$\frac{a}{b}=\frac{(3+2\sqrt{2})}{(3-2\sqrt{2})}$

Thus, the ratio is $( 3+ 2 \sqrt 2 ) : ( 3 - 2 \sqrt 2 )$

Question:29 If A and G be A.M. and G.M., respectively between two positive numbers, prove that the numbers are $A \pm \sqrt{( A+G)(A-G)}$

Answer:

If A and G be A.M. and G.M., respectively between two positive numbers,
Two numbers be a and b.

$AM=A=\frac{a+b}{2}$

$\Rightarrow a+b=2A$ ...................................................................1

$GM=G=\sqrt{ab}$

$\Rightarrow ab=G^2$ ...........................................................................2

We know $(a-b)^2=(a+b)^2-4ab$

Put values from equation 1 and 2,

$(a-b)^2=4A^2-4G^2$

$(a-b)^2=4(A^2-G^2)$

$(a-b)^2=4(A+G)(A-G)$

$(a-b)=4\sqrt{(A+G)(A-G)}$ ..................................................................3

From 1 and 3 , we have

$2a=2A+2\sqrt{(A+G)(A-G)}$

$\Rightarrow a=A+\sqrt{(A+G)(A-G)}$

Put value of a in equation 1, we get

$b=2A-A-\sqrt{(A+G)(A-G)}$

$\Rightarrow b=A-\sqrt{(A+G)(A-G)}$

Thus, numbers are $A \pm \sqrt{( A+G)(A-G)}$

Question:30 The number of bacteria in a certain culture doubles every hour. If there were 30 bacteria present in the culture originally, how many bacteria will be present at the end of 2nd hour, 4th hour and nth hour ?

Answer:

The number of bacteria in a certain culture doubles every hour.It forms GP.

Given : a=30 and r=2.

$a_3=a.r^{3-1}=30(2)^2=120$

$a_5=a.r^{5-1}=30(2)^4=480$

$a_n+_1=a.r^{n+1-1}=30(2)^n$

Thus, bacteria present at the end of the 2nd hour, 4th hour and nth hour are 120,480 and $30(2)^n$ respectively.

Question:31 What will Rs 500 amounts to in 10 years after its deposit in a bank which pays annual interest rate of 10% compounded annually?

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

$=500\left ( 1+\frac{1}{10} \right )=500(1.1)$

At the end of the second year, the amount $=500(1.1)(1.1)$

At the end of the third year, the amount $=500(1.1)(1.1)(1.1)$

At the end of 10 years, the amount $=500(1.1)(1.1)(1.1)........(10times)$

$=500(1.1)^{10}$

Thus, at the end of 10 years, amount $=Rs. 500(1.1)^{10}$

Question:32 If A.M. and G.M. of roots of a quadratic equation are 8 and 5, respectively, then obtain the quadratic equation

Answer:

Let roots of the quadratic equation be a and b.

According to given condition,

$AM=\frac{a+b}{2}=8$

$\Rightarrow (a+b)=16$

$GM=\sqrt{ab}=5$

$\Rightarrow ab=25$

We know that $x^2-x(sum\, of\, roots)+(product\, of\, roots)=0$

$x^2-x(16)+(25)=0$

$x^2-16x+25=0$

Thus, the quadratic equation = $x^2-16x+25=0$

Benefits of NCERT Solutions for Class 11 Maths Chapter 9 Exercise 9.3:-

NCERT book Class 11 maths chapter 9 exercise 9.3 is more lengthy and many concepts of progression are covered in it.
Most of the questions in exercise 9.3 Class 11 Maths are conceptually based on the geometric progression and arithmetic progression concepts.
You must try to solve all the problems of this exercise on your own before moving to the NCERT solutions for Class 11 Maths chapter 9 exercise 9.3.

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Key Features of NCERT 11th Class Maths Exercise 9.3 Answers

Comprehensive Coverage: The answers encompass all the ex 9.3 class 11 problem of the NCERT 11th Class Mathematics textbook.
Step-by-Step Solutions: The class 11 maths ex 9.3 solutions are presented in a step-by-step format, making it easier for students to understand and follow the problem-solving process.
Clarity and Precision: The class 11 ex 9.3 answers are written with clarity and precision, ensuring that students can grasp the mathematical concepts and methods required for effective problem-solving.
Conceptual Understanding: The 11th class maths exercise 9.3 answers aim to promote a deep understanding of mathematical concepts rather than rote memorization, encouraging critical thinking and problem-solving skills.
Free Accessibility: Typically, these class 11 maths ex 9.3 are available free of charge, allowing students to access them without any cost, making it a valuable resource for self-study.
Supplementary Learning Resource: These ex 9.3 class 11 answers can be used as a supplementary learning resource to reinforce classroom instruction and aid in exam preparation.
Homework and Practice: Students can utilize these answers to verify their work, practice problem-solving, and enhance their overall performance in mathematics.

Also see-

NCERT Solutions of Class 11 Subject Wise

Subject Wise NCERT Exampler Solutions

Happy learning!!!

Frequently Asked Question (FAQs)

1. Find the 5th term of G.P. 2,4,8...

Given a = 2

r = 4/2 = 2

a_5 = ar^4 = 2(2)^4 = 32

2. If the first term and third term of the G.P. are 3, 48 respectively than find the common ratio of the G.P. ?

Given a = 3

ar^2 = 48

3(r^2) = 48

r^2 = 16

Common ratio (r) = 4

3. Find the arithmetic mean of two positive integers a and b ?

A.M. = (a+b)/2

4. Find the geometric mean of two positive integers a and b ?

G.M. = (ab)^(1/2)

5. What is the sum of first n natural numbers?

Sum of first n natural numbers = n(n+1)/2

6. What is the sum of square of first n natural numbers?

Sum of square of first n natural numbers = n(n+1)(2n+1)/6

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Option 1) $0.34\; J$	Option 2) $0.16\; J$
Option 3) $1.00\; J$	Option 4) $0.67\; J$

Option 1) $2,000 \; J - 5,000\; J$	Option 2) $200 \, \, J - 500 \, \, J$
Option 3) $2\times 10^{5}J-3\times 10^{5}J$	Option 4) $20,000 \, \, J - 50,000 \, \, J$

Option 1) $K/2\,$	Option 2) $\; K\;$
Option 3) $zero\;$	Option 4) $K/4$

Option 1) $11.2\, L\, H_{2(g)}$ at STP is produced for every mole $HCL_{(aq)}$ consumed	Option 2) $6L\, HCl_{(aq)}$ is consumed for ever $3L\, H_{2(g)}$ produced
Option 3) $33.6 L\, H_{2(g)}$ is produced regardless of temperature and pressure for every mole Al that reacts	Option 4) $67.2\, L\, H_{2(g)}$ at STP is produced for every mole Al that reacts .

Option 1) 0.02	Option 2) 3.125 × 10^-2
Option 3) 1.25 × 10^-2	Option 4) 2.5 × 10^-2

Option 1) decrease twice	Option 2) increase two fold
Option 3) remain unchanged	Option 4) be a function of the molecular mass of the substance.

Option 1) Molality	Option 2) Weight fraction of solute
Option 3) Fraction of solute present in water	Option 4) Mole fraction.

Option 1) twice that in 60 g carbon	Option 2) 6.023 × 10²²
Option 3) half that in 8 g He	Option 4) 558.5 × 6.023 × 10²³

Option 1) less than 3	Option 2) more than 3 but less than 6
Option 3) more than 6 but less than 9	Option 4) more than 9

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NCERT Solutions for Exercise 9.3 Class 11 Maths Chapter 9 - Sequences and Series

NCERT Solutions for Class 11 Maths Chapter 9: Sequences and Series Exercise 9.3- Download Free PDF

Download the PDF of NCERT Solutions for Class 11 Maths Chapter 9: Sequences and Series Exercise 9.3

Access Sequences And Series Class 11 Chapter 9 Exercise 9.3

Question:1 Find the and terms of the G.P.

More About NCERT Solutions for Class 11 Maths Chapter 9 Exercise 9.3:-

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Question:1 Find the $20 ^{th}$ and $n ^{th}$ terms of the G.P. $\frac{5}{2},\frac{5}{4},\frac{5}{8},....$