If the sum of first 12 terms of series is 200, if the common difference is 4 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, ${\text{S}}_{12}=200$, n = 12, d = 4

Therefore,

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

${\text{S}}_{12}=\left(\frac{12}{2}\right)\left[2a+\left(12-1\right)4\right]$

$200=6\left[2a+\left(11\right)4\right]$

$200=6\left(2a+44\right)$

$200=12a+264$

$12a=200-264$

$12a=-64$

$a=\frac{-64}{12}=-5.3$

$a=-5.3$

Hence, the first term is -5.3.