Sum of Infinite Terms of a GP

Q. The sum of first three terms of a G.P. is $16$ and the sum of its next three terms is $128.$ Then the sum of next three terms of G.P is

1. $\frac{2008}{7}$
2. $\frac{1604}{7}$
3. $\frac{4016}{7}$
4. $1024$

Q. If $\alpha$ and $\beta$ are the roots of the equation $375x^2-25x-2=0,$ then $\underset{n\to \infty }{lim}\sum _{r=1}^n{\alpha }^r+\underset{n\to \infty }{lim}\sum _{r=1}^n{\beta }^r$ is equal to :

1. $\frac{7}{116}$
2. $\frac{1}{12}$
3. $\frac{29}{358}$
4. $\frac{21}{346}$

Q. If $A_i$ is the area bounded by $|x-a_i|+|y|=b_i,$ where $a_{i+1}=a_i+\frac{3}{2}{b}_i$ and $b_{i+1}=\frac{b_i}{2};$ $a_1=0,b_1=32,$ then

1. $A_3=128$
2. $A_3=256$
3. $\underset{n\to \infty }{lim}\sum _{i=1}^n{A}_i=\frac{8}{3}(32{)}^2$
4. $\underset{n\to \infty }{lim}\sum _{i=1}^n{A}_i=\frac{4}{3}(16{)}^2$

Q. The value of $6^{1/2}\times 6^{1/4}\times 6^{1/8}\times \cdots \infty$ is

Q.

Find a formula for the general term an of the sequence, assuming that the pattern of the first few terms continues. (Assume that n begins with 1.) $-5,\frac{10}{3},\frac{-20}{9},\frac{40}{27},\frac{-80}{81},...$

Q. The values of $x$ for which the function $f\left(x\right)=\frac{x^2+x+1}{x^2-7x+12}$ is not defined, is

1. $\{3,4\}$
2. $\{3,-4\}$
3. $\{-3,-4\}$
4. the function is defined everywhere

Q. The sum of $(11{)}^2+(12{)}^2+(13{)}^2+\dots +(20{)}^2$ is

1. $3485$
2. $2555$
3. $2885$
4. $2485$

Q. Consider an infinite geometric series with first term $a$ and common ratio $r$. If its sum is $4$ and the second term is $\frac{3}{4}$, then

1. $a=\frac{4}{7},r=\frac{3}{7}$
2. $a=2,r=\frac{3}{8}$
3. $a=\frac{3}{2},r=\frac{1}{2}$
4. $a=3,r=\frac{1}{4}$

Q. Let the $n^{th}$ terms of an A.P., G.P. and H.P. be $a,b,c$ respectively. If the first and the $(2n-1{)}^{th}$ terms of the A.P., G.P. and H.P. are equal, then which of the following is/are correct?

1. $a\le b\le c$
2. $a\ge b\ge c$
3. $bc-a^2=0$
4. $ac-b^2=0$

Q.

For each n ∈ N, the correct statement is

1. $2^n$ < n

2. $n^2$ > 2n

3. $n^4$ < ${10}^n$

4. $2^{3n}$ > 7n + 1

Q. If $S_{2n}=3S_n$ and $S_{5n}=k{S}_{3n}$, where $S_n$ is the sum of $n$ terms of an A.P., then the value of $k$ is

1. $2$
2. $2.5$
3. $3$
4. $3.5$

Q. Let $a_n$ denote the $n^{\text{th}}$ term of a G.P. If $a_1=3,a_n=96$ and sum of $n$ terms of the series is $189$, then the value of $n$ is

Q. The side length of the square whose area is equal to the $7^{th}$ term of the sequence $2,\sqrt{8},4,\dots ,$ is

1. $3$
2. $4$
3. $5$
4. $6$

Q. If each term of an infinite G.P. is twice the sum of the terms following it, then the common ratio of G.P. is

1. $\frac{1}{2}$
2. $\frac{1}{3}$
3. $\frac{1}{4}$
4. $\frac{1}{5}$

Q.

$x=1+a+a^2+...\infty (a<1)$
$y=1+b+b^2+...\infty (b<1)$
Then the value of $1+ab+a^2{b}^2+.....\infty$ is
[MNR 1980; MP PET 1985]

1. $\frac{xy}{x+y-1}$
2. $\frac{xy}{x-y+1}$
3. $\frac{xy}{x+y+1}$
4. $\frac{xy}{x-y-1}$

Q. The sum of the series $S=1+2\left(\frac{10}{11}\right)+3{\left(\frac{10}{11}\right)}^2+\cdots \text{upto}\infty$ is equal to

1. $121$
2. $120$
3. $111$
4. $110$

Q. Let $a,b,c$ are in AP and $a^2,b^2,c^2$ are in G.P. If $a<b<c$ and $a+b+c=\frac{3}{2}$ then $a=$

1. $\frac{1}{2}-\frac{1}{2\sqrt{2}}$
2. $\frac{1}{2}-\frac{1}{\sqrt{2}}$
3. $\frac{1}{2}+\frac{1}{\sqrt{2}}$
4. $\frac{1}{2}$

Q. If $b$ is the first term of an infinite $G.P.$ whose sum is five, then $b$ lies in the interval :

1. $(-\infty ,-10]$
2. $(-10,0)$
3. $(0,10)$
4. $[10,\infty )$

Q. Let $a,b,c$ are in AP and $a^2,b^2,c^2$ are in G.P. If $a<b<c$ and $a+b+c=\frac{3}{2}$ then $a=$

1. $\frac{1}{2}+\frac{1}{\sqrt{2}}$
2. $\frac{1}{2}$
3. $\frac{1}{2}-\frac{1}{2\sqrt{2}}$
4. $\frac{1}{2}-\frac{1}{\sqrt{2}}$

Q. If $\alpha$ and $\beta$ are the roots of the equation $375x^2-25x-2=0,$ then $\underset{n\to \infty }{lim}\sum _{r=1}^n{\alpha }^r+\underset{n\to \infty }{lim}\sum _{r=1}^n{\beta }^r$ is equal to :

1. $\frac{7}{116}$
2. $\frac{1}{12}$
3. $\frac{29}{358}$
4. $\frac{21}{346}$

Q. $\sqrt{-1-\sqrt{-1-\sqrt{-1-\cdots \text{to}\infty }}}$ is equal to :

1. ${\omega }^2$
2. $-\omega$
3. $1$
4. $-1$

Q. If $x=\sum _{n=0}^{\infty }(-1{)}^n{\tan }^{2n}\theta$ and $y=\sum _{n=0}^{\infty }{\cos }^{2n}\theta ,$ where $0<\theta <\frac{\pi }{4},$ then:

1. $y(1+x)=1$
2. $x(1-y)=1$
3. $y(1-x)=1$
4. $x(1+y)=1$

Q. After striking the floor ball rebounds ${\frac{4}{5}}^{\text{th}}$ of its height from which it has fallen. If it is released from a height of $120$m, then the total distance travelled by the ball (in $\text{m}$) before it comes to rest is

1. $960$
2. $1000$
3. $1080$
4. $2040$

Q. The product $2^{\frac{1}{4}}\times 4^{\frac{1}{16}}\times 8^{\frac{1}{48}}\times {16}^{\frac{1}{128}}...$ to $\infty$ is equal to :

1. $2^{\frac{1}{4}}$
2. $2$
3. $2^{\frac{1}{2}}$
4. $1$

Q. The sum of the first three terms of a G.P. is $6$ and the sum of its first three odd terms is $10.5$. Then the sum of its first term and the common ratio can be

1. $\frac{17}{12}$
2. $\frac{15}{2}$
3. $\frac{105}{38}$
4. $\frac{115}{38}$

Q. The product $2^{\frac{1}{4}}\times 4^{\frac{1}{16}}\times 8^{\frac{1}{48}}\times {16}^{\frac{1}{128}}...$ to $\infty$ is equal to :

1. $2^{\frac{1}{4}}$
2. $2$
3. $2^{\frac{1}{2}}$
4. $1$