If the sum of first 20 terms of series is 204, if the first term is 4 what will be the common difference?

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, $S_{20}=204,$ n = 20, a = 4

Therefore,

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

${\text{S}}_{20}=\left(\frac{20}{2}\right)\left[2a+\left(20-1\right)d\right]$

$204=10\left[2\left(4\right)+\left(19\right)d\right]$

$204=10\left(8+19d\right)$

$204=80+190d$

$190d=204-80$

$d=\frac{124}{190}=0.65$

$d=0.65$

Hence, the common difference is 0.65.