What is the arithmetic mean of the series $a + (a + d) + (a + 2d) + \dots + (a + 2nd)$?

Let the sequence $a_{n}$ be defined as follows:
$a_{1} = 1, a_{n} = a_{n-1} + 2$ for $n \ge 2$
Find the first five terms and write the corresponding series.

Let $S_{1}$ be the sum of the first $2n$ terms of an arithmetic progression. Let $S_{2}$ be the sum of the first $4n$ terms of the same arithmetic progression. If $(S_{2} - S_{1})$ is $1000$,then the sum of the first $6n$ terms of the arithmetic progression is equal to:

If the sum of the first $2n$ terms of the arithmetic progression $2, 5, 8, \dots$ is equal to the sum of the first $n$ terms of the arithmetic progression $57, 59, 61, \dots$,then $n = \dots$

If the sum of $n$ terms of an $A.P.$ is $2n^2 + 5n$,then the $n^{th}$ term will be

If $\log 2$,$\log (2^x - 1)$,and $\log (2^x + 3)$ are in arithmetic progression,then the value of $x$ is: