The value of $x$ satisfying $\log_a x + \log_{\sqrt{a}} x + \log_{\sqrt[3]{a}} x + \dots + \log_{\sqrt[a]{a}} x = \frac{a(a+1)}{2}$ is:

If $a_m$ denotes the $m^{th}$ term of an $A.P.$,then $a_m$ =

If $T_n = (n^2 + 1)n!$ and $S_n = T_1 + T_2 + T_3 + ...... + T_n$. Let $\frac{T_{10}}{S_{10}} = \frac{a}{b}$,where $a$ and $b$ are relatively prime natural numbers,then the value of $(b - a)$ is:

If the $9^{th}$ term of an $A.P.$ is zero,then the ratio of its $29^{th}$ term to its $19^{th}$ term is:

The sum of a $G.P.$ with common ratio $3$ is $364$, and the last term is $243$. Find the number of terms.

The sum of the first $n$ terms of the series $1^2 + 2.2^2 + 3^2 + 2.4^2 + 5^2 + 2.6^2 + \dots$ is $\frac{n(n + 1)^2}{2}$ when $n$ is even. When $n$ is odd,the sum will be: