If the sum of first ${}^{\text{'}}n^{\text{'}}$ natural numbers is $\frac{n(n+1)}{2}$, then what will be the sum of first $n$ terms of the series of alternate positive and negative numbers when ${}^{\text{'}}n^{\text{'}}$ is even?

$1^2-2^2+3^2-4^2+5^2-\dots$

$\mathrm{I}.$ $\frac{n(n+1)}{2}$

$\mathrm{II}.$ $\frac{n^2(n+1)}{2}$

$\mathrm{III}.$ $\frac{-n(n+1)}{2}$

Which of the above statement(s) is / are correct?

If general term of a G.P. with common ratio ${}^{\text{'}}r^{\text{'}}$ be $t_{k+1}$ and general term of another G.P. with common ratio ${}^{\text{'}}r^{\text{'}}$ be $t_{k+1}^{\text{'}}$, then the series whose general term $t_{k+1}^{\text{'}\text{'}}=t_{k+1}+t_{k+1}^{\text{'}}$ is

$\mathrm{I}.$ An A.P. with common difference ${}^{\text{'}}r^{\text{'}}$

$\mathrm{II}.$ A G.P. with common ratio ${}^{\text{'}}r^{\text{'}}$

$\mathrm{III}.$ A H.P.

Which of the above statement(s) is/are correct?

Let ${}^{\text{'}}a^{\text{'}}$ be the first term, ${}^{\text{'}}d^{\text{'}}$ be the common difference, $S_n$ denotes the sum of ${}^{\text{'}}n^{\text{'}}$ terms and ${{}^{\text{'}}a_n}^{\text{'}}$ denotes the $n^{\mathrm{th}}$ term of an A.P., then $\frac{S_m}{S_n}$ equals to $\frac{m^2}{n^2}$, when $m\ne n$ for some $m, n\in N$. The ratio $a_m: a_n$ is

$\mathrm{I}.$$\frac{m-1}{n-1}$

$\mathrm{II}.$ $\frac{2m-1}{2n-1}$

$\mathrm{III}.$ $\frac{{\sqrt{4S}}_m-\sqrt{a}}{\sqrt{4S_n}-\sqrt{a}}$

Which of the above statement(s) is / are correct?