Infinite GP

Q. The sum of an infinite number of terms of a G.P. is $20$, and the sum of their squares is $100$, then the common ratio is

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

Q. If $(1-y)(1+2x+4x^2+8x^3+16x^4+32x^5)=1-y^6,$ where $y\ne 1,x\ne \frac{1}{2}$, then the value of $\frac{y}{x}$ is

1. $\frac{25}{24}$
2. $\frac{1}{2}$
3. $2$
4. $\frac{24}{25}$

Q. If the ratio of $m^{th}$ and $n^{th}$ terms of an A.P. is $2m-1:2n-1$, then the ratio of sum of $m^{th}$ and $n^{th}$ terms of the series is

1. $m:n$
2. $m^2:n^2$
3. $\sqrt{m}:\sqrt{n}$
4. $m^2-1:n^2-1$

Q. Let $S\subset (0,\pi )$ denotes the set of values of $x$ satisfying the equation $8^{1+|\cos x|+{\cos }^{2}x+|{\cos }^{3}x|+\cdots +\infty }=4^3$. Then, $S=$

1. $\{\frac{\pi }{3}\}$
2. $\{-\frac{\pi }{3},\frac{2\pi }{3}\}$
3. $\{\frac{\pi }{3},\frac{2\pi }{3}\}$
4. $\{\frac{\pi }{3},-\frac{2\pi }{3}\}$

Q. If $x,y,z$ are in G.P. and $x+y,y+z,z+x$ are in A.P., where $x\ne y\ne z$, then common ratio of the G.P. is

1. $4$
2. $-1$
3. $2$
4. $-2$

Q. The sum of $(1+x)+(1+x+x^2)+(1+x+x^2+x^3)+\dots$ upto $n$ terms is

1. $\frac{1}{1-x}[n-\frac{x^2(1-x^n)}{1-x}]$
2. $\frac{1}{1-x}[n-\frac{x^3(1-x^n)}{1-x}]$
3. $\frac{1}{1-x}[n-\frac{x(1-x^n)}{1-x}]$
4. $\frac{1}{1-x}[n-\frac{2x(1-x^n)}{1-x}]$

Q. If the product of $3$ consecutive numbers in G.P. is $216$ and the sum of their products in pairs is $156$, then the smallest term in the three is

1. $2$
2. $6$
3. $\frac{1}{4}$
4. $\frac{1}{2}$

Q. The least value of n ( a natural number), for which the sum S of the series $1+\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}+\cdots$ differs from $S_n$ by a quantity $<{10}^{-6}$ is

1. 21
2. 20
3. 19
4. None

Q. The sum of the series $(\sqrt{2}+1)+1+(\sqrt{2}-1)+\cdots$ up to infinite term is

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

Q.

If sum of infinite terms of a G.P. is 3 and sum of squares of its terms is 3, then its first term and common ratio are
[RPET 1999]

1. 3/2, 1/2

2. 1, 1/2

3. 3/2, 2

4. None of these

Q. Let $S\subset (0,\pi )$ denotes the set of values of $x$ satisfying the equation $8^{1+|\cos x|+{\cos }^{2}x+|{\cos }^{3}x|+\cdots +\infty }=4^3$. Then, $S=$

1. $\{\frac{\pi }{3}\}$
2. $\{\frac{\pi }{3},-\frac{2\pi }{3}\}$
3. $\{-\frac{\pi }{3},\frac{2\pi }{3}\}$
4. $\{\frac{\pi }{3},\frac{2\pi }{3}\}$

Q. Sum of infinite number of terms in G.P. is 20 and sum of their square is 100. The common ratio of G.P. is
[AIEEE 2002]

1. 5

2. 3/5

3. 8/5
4. 1/5