NCERT Exemplar Class 11 Maths solutions Chapter 9 Sequence and Series

NCERT Exemplar Class 11 Maths Solutions Chapter 9- Excercise: 1.3

Question:1

The first term of an A.P.is a, and the sum of the first p terms is zero, show that the sum of its next q terms is $\frac{-a(p+q)q}{p-1}$.
[Hint: Required sum $S\text{textsubscript}p+q-S\text{textsubscript}p$]

Answer:

The sum of first p terms and the first term is given. We have to find next q terms. So, the total terms become p+q.
Hence, sum of all terms minus the sum of first p terms will give the sum of next q terms. However, sum of first p terms is zero, so the sum of next q terms will be same as sum of all terms.
Sum of n terms is given by $S_n=\left(\frac{n}{2}\right)(2a+(n-1)d)$
$$\begin{gathered} \text{ required sum }=S_{p+q}-S_p=\frac{p+q}{2}(2a+(p+q-1)d)\dots \dots (i) \\ {S}_p=\frac{p}{2}(2a+(p-1)d \\ 0=2a+(p-1)d \\ d=-\frac{2a}{p-1} \end{gathered}$$

Replacing the value of d in equation (i)

$$\begin{gathered} \text{required sum}=\frac{p+q}{2}(2a+(p+q-1)(-\frac{2a}{p-1}) \\ =\frac{p+q}{2}(2a-\frac{2ap+2aq-2a}{p-1}) \\ =\frac{p+q}{2}(2a)(1-\frac{p-1}{p-1}-\frac{q}{p-1}) \\ \text{On simplifying,} \\ =a(p+q)(-\frac{q}{p-1})=-\frac{a(p+q)q}{p-1} \end{gathered}$$

Question:2

A man saved Rs. 66000 in 20 years. In each succeeding year after the first year he saved Rs 200 more than what he saved in the previous year. How much did he save in the first year?

Answer:

Total amount saved in 20 years is $S_{20}=66000$
Let the amount saved in first year be ‘a’
Since he increases 200 Rs every year, d=200
Sum formula of A.P is $S_n=\left(\frac{n}{2}\right)(2a+(n-1)d)$

$S_{20}=\left(\frac{20}{2}\right)(2a+(20-1)200)$
$66000=10(2a+3800)$
$a=1400$

Question:3

A man accepts a position with an initial salary of Rs 5200 per month. It is understood that he will receive an automatic increase of Rs 320 in the very next month and each month thereafter.
(a) Find his salary for the tenth month
(b) What is his total earnings during the first year?

Answer:

The man’s salary in first month is Rs. 5200 and increases every month by Rs. 320. So, we have the value of ‘a’ and ‘d’.

$t_n=a+(n-1)d$

Salary in 10^th month is given by $t_{10}=5200+(10-1)320=8080$

For total earnings in first year, the first 12 terms of the sequence are added.

$S_n=\left(\frac{n}{2}\right)(2a+(n-1)d)$

$S_{12}=\left(\frac{12}{2}\right)(2\left(5200\right)+(12-1)320)$

Question:4

If the $p\text{textsuperscript}thandq\text{textsuperscript}th$ terms of a G.P. are q and p respectively, show that its $(p+q)\text{textsuperscript}th$ term is ${\left(\frac{q^p}{p^4}\right)}^{\frac{1}{p-8}}$

Answer:

The nth term is given by $t_n=a{r}^{n-1}$
The pth term is given by $t_p=a{r}^{p-1}$
$q=a{r}^{p-1}$
$\frac{q}{r^p}=\frac{a}{r}$
$t_q=a{r}^{q-1}$
$p=a{r}^{q-1}$
$$\begin{gathered} \frac{p}{r^q}=\frac{a}{r} \\ \frac{p}{r^q}=\frac{q}{r^p} \\ {r}^{p-q}=\frac{q}{p} \end{gathered}$$
$r={\left(\frac{q}{p}\right)}^{\frac{1}{p-q}}$
$t_{p+q}=a{\left(r\right)}^{p+q-1}$
$$\begin{gathered} t_{p+q}=\left(a{r}^{p-1}\right)r^q \\ {t}_{p+q}=q{r}^q \\ {t}_{p+q}=q{\left({\left(\frac{q}{p}\right)}^{\frac{1}{p-q}}\right)}^q \end{gathered}$$
$t_{p+q}=q\left(\frac{q^{\frac{q}{p-q}}}{p^{\frac{q}{p-q}}}\right)=\left(\frac{q^{1+\frac{q}{p-q}}}{p^{\frac{q}{p-q}}}\right)=\left(\frac{q^{\frac{p}{p-q}}}{p^{\frac{q}{p-q}}}\right)={\left(\frac{q^p}{p^q}\right)}^{\frac{1}{p-q}}$

Question:5

A carpenter was hired to build 192 window frames. The first day he made five frames and each day, thereafter he made two more frames than he made the day before. How many days did it take him to finish the job?

Answer:

The first day he made 5 frames and 2 more frames each passing day. So,
a=5 and d=2
$S_n=\left(\frac{n}{2}\right)(2a+(n-1)d)$

Let us assume that it takes ‘n’ days to build 192 window frames

$$\begin{gathered} {S}_n=\left(\frac{n}{2}\right)(10+(n-1)2) \\ 192=\left(\frac{n}{2}\right)(8+2n) \\ 384=2n^2+8n \\ {n}^2+4n-192=0 \\ (n-12)(n+16)=0 \end{gathered}$$

Since number of days can’t be negative n=12

Question:6

We know the sum of the interior angles of a triangle is 180°. Show that the sums of the interior angles of polygons with 3, 4, 5, 6, ... sides form an arithmetic progression. Find the sum of the interior angles for a 21-sided polygon.

Answer:

The sum of interir angles of a polygon having 'n' sides is given by $(n-2)\ast {180}^{\circ }$
By this formula sum of angles with 3 sides, 4 sides, 5 sides and 6 sides is ${180}^{\circ },{360}^{\circ },{540}^{\circ },{720}^{\circ }$
respectively. As the number of side increases by 1, the sum of interior angle increases by ${180}^{\circ }$
So,for sum of angles of polygon with 21 sides $=(n-2)\ast {180}^{\circ }$
$$\begin{gathered} =(21-2)\ast {180}^{\circ }=19\ast {180}^{\circ } \\ ={3420}^{\circ } \end{gathered}$$

Question:7

A side of an equilateral triangle is 20cm long. A second equilateral triangle is inscribed in it by joining the mid points of the sides of the first triangle. The process is continued as shown in the accompanying diagram. Find the perimeter of the sixth inscribed equilateral triangle.

Answer:

Let AB=BC=AC=20cm
Let D, E and F be midpoints of AC, CB and AB which are joined to form an equilateral triangle DEF. So, CD=CE=10cm. Triangle CDE is equilateral. Hence, DE= 10 cm. Similarly, GH = 5 cm
The series of sides of equilateral triangle will be 20,10,5…..
The series is a G.P with first term 20 and common ratio =1/2
$t_n=20{\left(\frac{1}{2}\right)}^{n-1}$

For the perimeter of 6^th triangle we first have to find the side of 6^th triangle

$t_6=\frac{20}{2^{6-1}}=\frac{20}{2^5}=\frac{5}{8}cm$

Perimeter would be thrice the length of its side.

Perimeter of triangle= $3\left(\frac{5}{8}\right)=\frac{15}{8}cm$

Question:8

In a potato race 20 potatoes are placed in a line at intervals of 4 metres with the first potato 24 metres from the starting point. A contestant is required to bring the potatoes back to the starting place one at a time. How far would he run in bringing back all the potatoes?

Answer:

It is given that at start he has to run 24m to get the first potato then 28 m as the next potato is 4m away and so on.
Hence, the sequence of its running will be 24, 28, 32…..
To get potatoes from the starting point he has to run 24+ 28+ 32+….. up to 20 terms
$$\begin{gathered} {S}_n=\left(\frac{n}{2}\right)(2a+(n-1)d) \\ {S}_{20}=\left(\frac{20}{2}\right)(2\left(24\right)+(20-1)4) \\ {S}_{20}=10(48+76) \\ {S}_{20}=1240m \end{gathered}$$

Also, he has to get the potato back to the starting point hence the total distance will be twice
Total Distance ran=2×1240=2480m

Question:9

In a cricket tournament 16 school teams participated. A sum of Rs 8000 is to be awarded among themselves as prize money. If the last placed team is awarded Rs 275 in prize money and the award increases by the same amount for successive finishing places, how much amount will the first place team receive?

Answer:

Let the amount received by first place be Rs. A and ‘d’ be the difference in the amount. The last team receives Rs. 275. As there are 16 teams and all teams are given prizes hence the sequence will have 16 terms because there are 16 teams.

As total prize given is of 8000 rs hence

$a+a-d+a-2d....+275=8000$

$$\begin{gathered} n=16 \\ {S}_{16}=8000 \\ {S}_n=\left(\frac{n}{2}\right)(2a+(n-1)(-d)) \\ 8000=\left(\frac{16}{2}\right)(2a+(16-1)(-d))=8(2a-15d) \\ 1000=2a-15d \\ {t}_{16}=275 \\ {t}_{16}=a+(16-1)(-d) \\ 275=a-15d \\ 1000-275=(2a-15d)-(a-15d) \\ a=725 \end{gathered}$$

Question:10

If $a\text{textsubscript}1,a\text{textsubscript}2,a\text{textsubscript}3,$ \ldots $,a\text{textsubscript}n$ are in A.P. $a\text{textsubscript}i>0$ for all i show that $\frac{1}{\sqrt{{\mathrm{a}}_1}+\sqrt{{\mathrm{a}}_2}}+\frac{1}{\sqrt{{\mathrm{a}}_2}+\sqrt{{\mathrm{a}}_3}}+\cdots +\frac{1}{\sqrt{{\mathrm{a}}_{\mathrm{n}-1}}+\sqrt{{\mathrm{a}}_{\mathrm{n}}}}=\frac{\mathrm{n}-1}{\sqrt{{\mathrm{a}}_1}+\sqrt{{\mathrm{a}}_{\mathrm{n}}}}$

Answer:

$LHS=\frac{1}{\sqrt{a_1}+\sqrt{a_2}}+\frac{1}{\sqrt{a_2}+\sqrt{a_3}}+\dots +\frac{1}{\sqrt{a_{n-1}}+\sqrt{a_n}}$
$\begin{array}{rl}& \text{Multiplying the first term by }\frac{\sqrt{a_1}-\sqrt{a_2}}{\sqrt{a_1}-\sqrt{a_2}},\text{ the second term by }\frac{\sqrt{a_2}-\sqrt{a_3}}{\sqrt{a_2}-\sqrt{a_3}}\\ & \text{and so on rationalising each term}\end{array}$

$LHS=\left(\frac{1}{\sqrt{a_1}+\sqrt{a_2}}\right)\left(\frac{\sqrt{a_1}-\sqrt{a_2}}{\sqrt{a_1}-\sqrt{a_2}}\right)+\left(\frac{1}{\sqrt{a_2}+\sqrt{a_3}}\right)\left(\frac{\sqrt{a_2}-\sqrt{a_3}}{\sqrt{a_2}-\sqrt{a_3}}\right)+\dots +\left(\frac{1}{\sqrt{a_{n-1}}+\sqrt{a_n}}\right)\left(\frac{\sqrt{a_{n-1}}-\sqrt{a_n}}{\sqrt{a_{n-1}}-\sqrt{a_n}}\right)$
$$\begin{gathered} =\left(\frac{\sqrt{a_1}-\sqrt{a_2}}{a_1-a_2}\right)+\left(\frac{\sqrt{a_2}-\sqrt{a_3}}{a_2-a_3}\right)+\dots +\left(\frac{\sqrt{a_{n-1}}-\sqrt{a_n}}{a_{n-1}-a_n}\right) \\ =\left(\frac{\sqrt{a_1}-\sqrt{a_2}}{-d}\right)+\left(\frac{\sqrt{a_2}-\sqrt{a_3}}{-d}\right)+\dots +\left(\frac{\sqrt{a_{n-1}}-\sqrt{a_n}}{-d}\right) \end{gathered}$$

$$\begin{gathered} =-\frac{1}{d}(\sqrt{a_1}-\sqrt{a_2}+\sqrt{a_2}-\sqrt{a_3}+\dots +\sqrt{a_{n-1}}-\sqrt{a_n}) \\ =-\frac{1}{d}(\sqrt{a_1}-\sqrt{a_n}) \end{gathered}$$

Rationalising again,

$$\begin{gathered} =-\frac{1}{d}(\sqrt{a_1}-\sqrt{a_n})\left(\frac{(\sqrt{a_1}+\sqrt{a_n})}{(\sqrt{a_1}+\sqrt{a_n})}\right) \\ =-\frac{1}{d}\frac{(a_1-a_n)}{(\sqrt{a_1}+\sqrt{a_n})}=-\frac{1}{d}\frac{(-(n-1)d)}{(\sqrt{a_1}+\sqrt{a_n})}=\frac{(n-1)}{(\sqrt{a_1}+\sqrt{a_n})} \end{gathered}$$

Question:11

Find the sum of the series to (i) n terms
(ii) 10 terms

Answer:

Generalising the series in terms of i
$$\begin{gathered} S=\sum _{i=1}^n[{(2i+1)}^3-{\left(2i\right)}^3] \\ =\sum _{i=1}^n\left[(2i+1-2i)({(2i+1)}^2+(2i+1)\left(2i\right)+{\left(2i\right)}^2)\right] \\ =\sum _{i=1}^n[4i^2+4i+1+4i^2+2i+4i^2]=\sum _{i=1}^n[12i^2+6i+1] \end{gathered}$$
$$\begin{gathered} =12\left[\frac{n(n+1)(2n+1)}{6}\right]+6\left[\frac{n(n+1)}{2}\right]+\left[n\right] \\ =2[2n^3+3n^2+n]+3[n^2+n]+\left[n\right] \end{gathered}$$
Sum up to n terms $=4n^3+9n^2+6n$
$\text{Sum up to 10 terms }{S}_{10}=4{\left(10\right)}^3+9{\left(10\right)}^2+6\left(10\right)=4960$

Question:12

Find the rth term of an A.P. sum of whose first n terms is $2n+3n^2$.
$[Hint:a\text{textsubscript}n=S\text{textsubscript}n-S\text{textsubscript}n-1]$

Answer:

Sum of first n terms is given by $S_n=2n+3n^2$
The rth term is
$$\begin{gathered} {a}_r=S_r-S_{r-1}=(2r+3r^2)-(2(r-1)+3{(r-1)}^2) \\ =2r+3r^2-(2r-2+3r^2-6r+3) \\ =2r+3r^2-(3r^2-4r+1) \\ =6r-1 \end{gathered}$$

Question:13

If A is the arithmetic mean and $G_1$, $G_2$ be two geometric means between any two numbers, then prove that
$2\mathrm{A}=\frac{{\mathrm{G}}_1^2}{{\mathrm{G}}_2}+\frac{{\mathrm{G}}_2^2}{{\mathrm{G}}_1}$

Answer:

It is given that A is the arithmetic mean and $G_1$, $G_2$ be two geometric means between any two numbers.
$$\begin{gathered} A=\frac{a+b}{2} \\ G=\sqrt{ab} \\ {G}_1=\sqrt{a{G}_2} \\ {G}_2=\sqrt{G_{1}b} \\ {G}_1^2=a{G}_2 \end{gathered}$$
$$\begin{gathered} {G}_1^2=a\sqrt{G_{1}b} \\ {G}_1^4=a^2{G}_{1}b \\ {G}_1^3=a^{2}b \\ {G}_1=a^{\frac{2}{3}}{b}^{\frac{1}{3}} \end{gathered}$$
$$\begin{gathered} {G}_2={(a^{\frac{2}{3}}{b}^{\frac{1}{3}}\times b)}^{\frac{1}{2}}=a^{\frac{1}{3}}{b}^{\frac{2}{3}} \\ \frac{G_1^2}{G_2}=\left(\frac{{\left(a^{\frac{2}{3}}{b}^{\frac{1}{3}}\right)}^2}{a^{\frac{1}{3}}{b}^{\frac{2}{3}}}\right)=a \\ \frac{G_2^2}{G_1}=\left(\frac{{\left(b^{\frac{2}{3}}{a}^{\frac{1}{3}}\right)}^2}{b^{\frac{1}{3}}{a}^{\frac{2}{3}}}\right)=b \\ \frac{G_1^2}{G_2}+\frac{G_2^2}{G_1}=a+b=2A \end{gathered}$$

Question:14

are in A.P., whose common difference is d, show that

Answer:

Given that ${\theta }_1,{\theta }_2,\dots {\theta }_n\text{~are in A.P and common difference is d. }$

$$\begin{gathered} \sec {\theta }_1\sec {\theta }_2+\sec {\theta }_2\sec {\theta }_3+\dots +\sec {\theta }_{n-1}\sec {\theta }_n \\ =\frac{1}{\cos {\theta }_1\cos {\theta }_2}+\frac{1}{\cos {\theta }_2\cos {\theta }_3}+\dots +\frac{1}{\cos {\theta }_{n-1}\cos {\theta }_n} \\ =\frac{\sin d}{\sin d}(\frac{1}{\cos {\theta }_1\cos {\theta }_2}+\frac{1}{\cos {\theta }_2\cos {\theta }_3}+\dots +\frac{1}{\cos {\theta }_{n-1}\cos {\theta }_n}) \end{gathered}$$
$$\begin{gathered} =\frac{1}{\sin d}(\frac{\sin ({\theta }_2-{\theta }_1)}{\cos {\theta }_1\cos {\theta }_2}+\frac{\sin ({\theta }_3-{\theta }_2)}{\cos {\theta }_2\cos {\theta }_3}+\dots +\frac{\sin ({\theta }_n-{\theta }_{n-1})}{\cos {\theta }_{n-1}\cos {\theta }_n}) \\ =\frac{1}{\sin d}(\frac{\sin {\theta }_2\cos {\theta }_1-\cos {\theta }_2\sin {\theta }_1}{\cos {\theta }_1\cos {\theta }_2}+\frac{\sin {\theta }_3\cos {\theta }_2-\cos {\theta }_3\sin {\theta }_2}{\cos {\theta }_2\cos {\theta }_3}+\dots +\frac{\sin {\theta }_n\cos {\theta }_{n-1}-\cos {\theta }_n\sin {\theta }_{n-1}}{\cos {\theta }_{n-1}\cos {\theta }_n}) \end{gathered}$$
$$\begin{gathered} =\frac{1}{\sin d}(\frac{\sin {\theta }_2\cos {\theta }_1}{\cos {\theta }_1\cos {\theta }_2}-\frac{\sin {\theta }_1\cos {\theta }_2}{\cos {\theta }_1\cos {\theta }_2}+\frac{\sin {\theta }_3\cos {\theta }_2}{\cos {\theta }_2\cos {\theta }_3}-\frac{\sin {\theta }_2\cos {\theta }_3}{\cos {\theta }_2\cos {\theta }_3}+\dots +\frac{\sin {\theta }_n\cos {\theta }_{n-1}}{\cos {\theta }_{n-1}\cos {\theta }_n}-\frac{\sin {\theta }_n\cos {\theta }_{n-1}}{\cos {\theta }_{n-1}\cos {\theta }_n}) \\ =\frac{1}{\sin d}(\tan {\theta }_2-\tan {\theta }_1+\tan {\theta }_3-\tan {\theta }_2+\dots +\tan {\theta }_n-\tan {\theta }_{n-1}) \end{gathered}$$

$=\frac{(\tan {\theta }_n-\tan {\theta }_1)}{\sin d}$

Question:15

If the sum of p terms of an A.P. is q and the sum of q terms is p, show that the sum of p + q terms is - (p + q). Also, find the sum of first p - q terms (p > q).

Answer:

The sum of n terms is given by
$S_n=\left(\frac{n}{2}\right)(2a+(n-1)d)$
Where ‘a’ is the first term and ’d’ is the common difference

$$\begin{gathered} {S}_p=q \\ q=\frac{p}{2}(2a+(p-1)d) \\ \frac{2q}{p}=2a+(p-1)d \\ {S}_q=p \\ p=\frac{q}{2}(2a+(q-1)d) \end{gathered}$$
$$\begin{gathered} \frac{2p}{q}=2a+(q-1)d \\ \frac{2p}{q}-\frac{2q}{p}=(q-1)d-(p-1)d \\ \frac{2p}{q}-\frac{2q}{p}=(q-p)d \\ d=\frac{2p^2-2q^2}{pq(q-p)}=-\frac{2(p+q)}{pq} \end{gathered}$$
$$\begin{gathered} {S}_{p+q}=\frac{p+q}{2}(2a+(p+q-1)d) \\ =\frac{p}{2}(2a+(p-1)d+qd)+\frac{q}{2}(2a+(q-1)d+pd) \end{gathered}$$
$$\begin{gathered} =\frac{p}{2}(2a+(p-1)d)+\frac{pqd}{2}+\frac{q}{2}(2a+(q-1)d)+\frac{pqd}{2} \\ =q+p+pqd \end{gathered}$$
$$\begin{gathered} =q+p+pq(-\frac{2(p+q)}{pq})=q+p-2p-2q \\ {S}_{p+q}=-(p+q) \\ {S}_{p-q}=\frac{p-q}{2}(2a+(p-q-1)d) \end{gathered}$$
$$\begin{gathered} =\frac{p}{2}(2a+(p-1)d)-\frac{pqd}{2}-\frac{q}{2}(2a+(p-1)d)+\frac{q^{2}d}{2} \\ =q-\frac{pqd}{2}-\left(\frac{q}{p}\right)\left(\frac{p}{2}\right)(2a+(p-1)d)+\frac{q^{2}d}{2} \\ =q-\frac{q^2}{p}-\frac{pqd}{2}+\frac{q^{2}d}{2} \\ =\frac{(pq-q^2)}{p}+\frac{d(q^2-pq)}{2} \\ =\frac{pq-q^2}{p}-\frac{pq-q^2}{2}(-\frac{2(p+q)}{pq}) \end{gathered}$$
$$\begin{gathered} =\frac{pq-q^2}{p}+(pq-q^2)[\frac{1}{q}+\frac{1}{p}] \\ =\frac{2(pq-q^2)}{p}+\frac{pq-q^2}{q} \\ =\frac{2q(p-q)}{p}+p-q \end{gathered}$$

Question:16

If $p^{th}$, $q^{th}$, and $r^{th}$ terms of an A.P. and G.P. are both a, b and c respectively, show that
$a\text{textsuperscript}b-c.b\text{textsuperscript}c-a.c\text{textsuperscript}a-b=1$

Answer:

Let the first term of AP be m and common difference as d. Let the first term of GP be I and common ratio be s.

For AP

$$\begin{gathered} {t}_p=m+(p-1)d \\ a=m+(p-1)d \\ b=m+(q-1)d \\ c=m+(r-1)d \end{gathered}$$

For GP
$$\begin{gathered} {t}_p=I{s}^{p-1} \\ a=I{s}^{p-1} \\ b=I{s}^{q-1} \\ c=I{s}^{r-1} \\ b-c=(q-r)d \end{gathered}$$

$$\begin{gathered} c-a=(r-p)d \\ a-b=(p-q)d \\ LHS=a^{b-c}{b}^{c-a}{c}^{a-b} \\ ={\left(I{s}^{p-1}\right)}^{(q-r)d}{\left(I{s}^{q-1}\right)}^{(r-p)d}{\left(I{s}^{r-1}\right)}^{(p-q)d} \\ =I^{(q-r+r-p+p-q)d}{s}^{[(pq-pr-q+r)+(qr-qp-r+p)+(rp-rq-p+q)]d} \\ =I^0{s}^0=1 \end{gathered}$$

Question:17

If the sum of n terms of an A.P. is given by $S_n=3n+2n^2$, then the common difference of the A.P. is
(A) 3 (B) 2 (C) 6 (D) 4

Answer:

Given that $S_n=3n+2n^2$
We know that $a_2=S_2-S_1$

$$\begin{gathered} =(3\left(2\right)+2{\left(2\right)}^2)-(3\left(1\right)+2{\left(1\right)}^2) \\ =14-5 \\ =9 \\ {a}_1=S_1=3\left(1\right)+2{\left(1\right)}^2=5 \\ d=a_2-a_1=9-5=4 \end{gathered}$$

Hence, correct option is d.

Question:18

The third term of G.P. is 4. The product of its first 5 terms is

(A) 4³ (B) 4⁴ (C) 4⁵ (D) None of these

Answer:

The third term of G.P is given $T_3=4$

$$\begin{gathered} {T}_n=a{r}^{n-1} \\ a{r}^2=4 \end{gathered}$$
Product of first 5 terms

$$\begin{gathered} =\left(a\right)\left(ar\right)\left(a{r}^2\right)\left(a{r}^3\right)\left(a{r}^4\right) \\ =\left(a^5{r}^{10}\right)={\left(a{r}^2\right)}^5 \\ =4^5=1024 \end{gathered}$$

Hence, correct option is c.

Question:19

If 9 times the 9th term of an A.P. is equal to 13 times the 13th term, then the 22nd term of the A.P. is
A. 0
B. 22
C. 220
D. 198

Answer:

$$\begin{gathered} 9a_9=13a_{13} \\ 9(a+(9-1)d)=13(a+(13-1)d) \\ 9a+72d=13a+156d \\ -4a=84d \\ a=-21d \\ {a}_{22}=a+(22-1)d \\ =-21d+21d=0 \end{gathered}$$

Hence, correct option is a.

Question:20

If x, 2y, 3z are in A.P., where the distinct numbers x, y, z are in G.P. then the common ratio of the G.P. is
A. 3
B. 1/3
C. 2
D. 1/2

Answer:

It is given that x, 2y and 3z are in A.P

$$\begin{gathered} 2y-x=3z-2y \\ x=4y-3z \\ \frac{y}{x}=\frac{z}{y} \\ {y}^2=xz \\ {y}^2=(4y-3z)z \\ {y}^2-4yz+3z^2=0 \\ (3z-y)(z-y)=0 \\ y=3zory=z \\ r=\frac{y}{z}=\frac{1}{3} \end{gathered}$$

Hence, correct option is b.

Question:21

If in an A.P., Sn = qn² and Sm = qm², where S_r denotes the sum of r terms of the A.P., then S_q equals
A. $\frac{q^3}{2}$
B. mnq
C. $q^3$
D. $(m+n)q^2$

Answer:

The given series is an A.P with first term ‘a’ and common difference ‘d’.

$$\begin{gathered} {S}_n=\frac{n}{2}[2a+(n-1)d] \\ q{n}^2=\frac{n}{2}[2a+(n-1)d] \\ 2qn=2a+(n-1)d \\ 2a=2qn-(n-1)d \\ {S}_m=\frac{n}{2}[2a+(m-1)d] \\ q{m}^2=\frac{m}{2}[2a+(m-1)d] \end{gathered}$$

$$\begin{gathered} 2qm=2a+(m-1)d \\ 2a=2qm-(m-1)d \\ 2qm-(m-1)d=2qn-(n-1)d \\ 2q(n-m)=(n-m)d \\ d=2q \\ 2a=2qn-(n-1)\left(2q\right) \end{gathered}$$

$$\begin{gathered} a=q \\ {S}_q=\frac{q}{2}[2q+(q-1)2q]=q^3 \end{gathered}$$

Hence, correct option is c.

Question:22

Let $S_n$ denote the sum of the first n terms of an A.P. If $S\text{textsubscript}2n=3S\text{textsubscript}n$ then $S\text{textsubscript}3n:S\text{textsubscript}n$ is equal to
A. 4
B. 6
C. 8
D. 10

Answer:

Given $S_{2n}=3S_n$

$$\begin{gathered} {S}_n=\frac{n}{2}[2a+(n-1)d] \\ {S}_{2n}=\frac{2n}{2}[2a+(2n-1)d] \\ {S}_{2n}=3S_n \\ n[2a+(2n-1)d]=\frac{3n}{2}[2a+(n-1)d] \\ 4a+4nd-2d=6a+3nd-3d \\ 2a=nd+d \end{gathered}$$

$$\begin{gathered} {S}_{3n}=\frac{3n}{2}[2a+(3n-1)d]=\frac{3n}{2}[(n+1)d+(3n-1)d]=6n^{2}d \\ {S}_n=\frac{n}{2}[2a+(n-1)d]=\frac{n}{2}[(n+1)d+(n-1)d]=n^{2}d \\ \frac{S_{3n}}{S_n}=\frac{6n^{2}d}{n^{2}d}=6 \end{gathered}$$

Hence, the correct option is b.

Question:23

The minimum value of $4\text{textsuperscript}x+4\text{textsuperscript}1-x,x\in R$ is
A. 2
B. 4
C. 1
D. 0

Answer:

The AM-GM inequality states that the arithmetic mean of a list of non-negative real numbers is greater than or equal to the geometric mean of the same list

$$\begin{gathered} \frac{x+y}{2}\ge \sqrt{xy} \\ \frac{4^x+4^{1-x}}{2}\ge \sqrt{\left(4^x\right)\left(4^{1-x}\right)} \\ \frac{4^x+4^{1-x}}{2}\ge \sqrt{4} \\ {4}^x+4^{1-x}\ge 4 \end{gathered}$$

Hence, correct option is b.

Question:24

Let $S_n$ denote the sum of the cubes of the first n natural numbers and $s_n$ denote the sum of the first n natural numbers. Then $\sum _{r=1}^n\frac{S_r}{s_r}$equals.
A. $\frac{\mathrm{n}(\mathrm{n}+1)(\mathrm{n}+2)}{6}$

B. $\frac{\mathrm{n}(\mathrm{n}+1)}{2}$
C. $\frac{{\mathrm{n}}^2+3\mathrm{n}+2}{2}$
D. None of these