If the sum of first 25 terms of series is 2250, if the common difference is 5 what will be the first term?

Concept: The sum of the first n terms in A.P. is given by:

$\Rightarrow {\text{S}}_n=\left(\frac{n}{2}\right)\left[2a+\left(n-1\right)d\right]$

Here, ${\text{S}}_{25}=2250$, n = 25, d = 5

Therefore,

${\text{S}}_n=\left(\frac{n}{2}\right)\left[2a+\left(n-1\right)d\right]$

${\text{S}}_{25}=\left(\frac{25}{2}\right)\left[2a+\left(25-1\right)5\right]$

$2250=\frac{25}{2}\left[2a+\left(24\right)5\right]$

$2250=12.5\left(2a+120\right)$

$2250=25a+1500$

$25a=750$

$a=\frac{750}{25}=30$

$a=30$

Hence, the first term is 30.