Sum of first $\text{'}n\text{'}$ terms of the sequence $5,7,11,17,25,\mathrm{.........}$  is equal to:

             $S=5+7+11+17+25+\mathrm{......}+T_n$

$\Rightarrow$        $S=5+7+11+17+\mathrm{........}+T_{n-1}+T_n$

$\Rightarrow$         $T_n=5+\left(2+4+6+8+\mathrm{......}to\left(n-1\right)terms\right)$

                   $=5+\left(n-1\right)n=n^2-n+5$

$\therefore$      $S=\frac{1}{6}.n\left(n+1\right)\left(2n+1\right)-\frac{1}{2}n\left(n+1\right)+5n$

$=\frac{n}{3}\left(n^2+14\right)$