Question

In an AP the sum of first n terms is $\frac{3n^2}{2}+\frac{5n}{2}$ Find ${25}^{th}$ term.

Answer 1

$S_n=\frac{3n^2+5n}{2}$

$t_n$ is given by

$t_n=S_n-S_{n-1}$

$t_{25}=S_{25}-S_{24}$

$S_{25}=\frac{3(25{)}^2+5(25)}{2}$

$=\frac{3\times 625+125}{2}$

$=\frac{1875+125}{2}$

$=\frac{2000}{2}=1000$

$S_{24}=\frac{3(24{)}^2+5(24)}{2}$

$=\frac{3\times 576+120}{2}=\frac{1728+120}{2}$

$=864+60=924$

$\therefore t_{25}=1000-924=76$

The ${25}^{th}$ term is $76$.