Question

The sum of the first n terms in an AP is ${\displaystyle (\frac{3{\text{n}}^2}{2}+\frac{5\text{n}}{2})}$. Find the n^th term and the 25^th term.

Answer 1

Let S  denotes the sum of first n terms of the AP.

∴ ${\displaystyle {\text{s}}_{\text{n}}=(\frac{3{\text{n}}^2}{2}+\frac{5\text{n}}{2})}$

⇒ ${\displaystyle {\text{s}}_{\text{n}-1}=\frac{3{(\text{n}-1)}^2}{2}+\frac{5(\text{n}-1)}{2}}$

${\displaystyle =\frac{3({\text{n}}^2-2\text{n}+1)}{2}+\frac{5(\text{n}-1)}{2}}$

${\displaystyle =\frac{3{\text{n}}^2-\text{n}-2}{2}}$

∴ n^th term pf the AP, a_n 

= ${\displaystyle {\text{s}}_{\text{n}}-{\text{s}}_{\text{n}-1}}$

= ${\displaystyle \left(\frac{3{\text{n}}^2+5\text{n}}{2}\right)-\left(\frac{3{\text{n}}^2-\text{n}-2}{2}\right)}$

= ${\displaystyle \frac{6\text{n}+2}{2}}$

= 3n + 2

Putting n = 25, we get

a₂₅ = 3 × 25 + 1 = 75 + 1 = 76

Hence, the n^th term is (3n + 2) and 25^th term is 76.