The sum of the first $n$ terms of a sequence is given by $S_n = 3n^2 + 4n + 15$. If $T_r$ is the $r^{th}$ term of the sequence,then $T_3 - T_1$ is equal to

If $A_1, A_2$ are two arithmetic means,$G_1, G_2$ are two geometric means,and $H_1, H_2$ are two harmonic means between two numbers,then $\frac{A_1 + A_2}{H_1 + H_2} \cdot \frac{H_1 H_2}{G_1 G_2} = \dots$

The number of common terms in the progressions $4, 9, 14, 19, \ldots$ up to $25^{\text{th}}$ term and $3, 6, 9, 12, \ldots$ up to $37^{\text{th}}$ term is:

Let the sum of an infinite $G.P.$,whose first term is $a$ and the common ratio is $r$,be $5$. Let the sum of its first five terms be $\frac{98}{25}$. Then the sum of the first $21$ terms of an $A.P.$,whose first term is $10ar$,$n^{\text{th}}$ term is $a_n$ and the common difference is $10ar^2$,is equal to.

$n^n \left( \frac{n+1}{2} \right)^{2n}$ is

Let $a_i = i + \frac{1}{i}$ for $i = 1, 2, \ldots, 20$. Let $p = \frac{1}{20} \sum_{i=1}^{20} a_i$ and $q = \frac{1}{20} \sum_{i=1}^{20} \frac{1}{a_i}$. Then,