If the sum of the first $n$ terms of a series be $5{n^2} + 2n$, then its second term is

If the sum of the series $54 + 51 + 48 + .............$ is $513$, then the number of terms are

Let the sequence $a_{n}$ be defined as follows:

${a_1} = 1,{a_n} = {a_{n - 1}} + 2$ for $n\, \ge \,2$

Find first five terms and write corresponding series.

If $3^{2 \sin 2 \alpha-1},14$ and $3^{4-2 \sin 2 \alpha}$ are the first three terms of an $A.P.$ for some $\alpha$, then the sixth term of this $A.P.$ is 

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

The sum of numbers from $250$ to $1000$ which are divisible by $3$ is