Question

The sum of $n$ terms of an $A.P$ is $3n^2+5n.$ Find the $A.P$ hence, find its ${15}^{th}$ term.

Answer 1

$\frac{n}{2}[2a+(n-1\left)d\right]=3n^2+5n$

$\frac{n}{2}(2a+(n-1\left)d\right]=n(3n+5)$

$2a+(n-1)d=6n+10$

$d=\frac{6n+10=2a}{(n-1)}$

${15}^{th}$ term

$\Rightarrow a+(n-1)d=a+14d$

$=a+14\times \left(\frac{6\times 15+10-2a}{14}\right)$

${15}^{th}$ term $=a+90+10-2a=100-a$.