If all the terms of an $A.P.$ are squared,then the new series will be in

Let $a_n = \frac{10^n}{n!}$ for $n = 1, 2, 3, \ldots$. Then the greatest value of $n$ for which $a_n$ is the greatest is

Let $a_1=b_1=1$ and $a_n=a_{n-1}+(n-1)$,$b_n=b_{n-1}+a_{n-1}$,$\forall n \geq 2$. If $S =\sum \limits_{n=1}^{10} \frac{b_n}{2^n}$ and $T =\sum \limits_{n=1}^8 \frac{n}{2^{n-1}}$,then $2^7(2S - T)$ is equal to $........$.

If the value of $\left(1+\frac{2}{3}+\frac{6}{3^{2}}+\frac{10}{3^{3}}+\ldots \text{ to } \infty\right)^{\log_{(0.25)}\left(\frac{1}{3}+\frac{1}{3^{2}}+\frac{1}{3^{3}}+\ldots \text{ to } \infty\right)}$ is $l$,then $l^{2}$ is equal to $......$

Let the sum of the first $n$ terms of an $A$.$P$. be $3n^2 + 5n$. Then the sum of squares of the first $10$ terms of the $A$.$P$. is:

If three unequal numbers $p, q, r$ are in $H.P.$ and their squares are in $A.P.$,then the ratio $p:q:r$ is