Infinite GP

Q.

If the variance of $10$ natural numbers $1,1,1,...,1,k$ is less than $10$, then the maximum possible value of $k$ is

Q.

The third term of a G.P is the square of first term. If the second term is $8$, then the $6^{th}$ term is

1. $120$

2. $124$

3. $128$

4. $132$

Q.

Find the $8^{\mathrm{th}}$ term of the GP : $0.3,0.06,0.012,\dots$

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. 19
2. 21
3. None
4. 20

Q.

If $S_1,S_2,S_3,\dots S_m$ are the sums of $n$ terms of $mA.P.$ are $n$ and $m$
respectively, then find the sum of common differences are $1,3,5,\dots ,(2m-1)$ respectively, show that
$S_1+S_2+S_3+\dots S_m=\frac{1}{2}nm(nm+1)$

Q.

The product of infinite terms in $x^{\frac{1}{2}}.x^{\frac{1}{4}}.x^{\frac{1}{8}}...........\infty$ is

1. 0

2. infinity

3. 1

4. x