Question

In an AP, the sum of first$n$Terms Is $\frac{3n^2}{2}+\frac{3n}{2}$ . find the ${25}^{th}$ term.

Answer 1

Find the ${25}^{th}$ term of given AP

Given : $S_{n=}=\frac{3n^2}{2}+\frac{3n}{2}$

So, the ${25}^{th}$ term of given AP $=S_{25}-S_{24}$

$$\begin{gathered} =\left[\frac{3{\left(25\right)}^2}{2}+\frac{3\left(25\right)}{2}\right]-\left[\frac{3{\left(24\right)}^2}{2}+\frac{3\left(24\right)}{2}\right] \\ =\left[\frac{1875}{2}+\frac{75}{2}\right]-\left[\frac{1728}{2}+\frac{72}{2}\right] \\ =\left[\frac{1950}{2}\right]-\left[\frac{1800}{2}\right] \\ =975-900 \\ =75 \end{gathered}$$

Hence, the ${25}^{th}$ term of given AP $75$.