The sum of the third and seventh terms; of an A.P. is 6 and their product is 8. Find the sum of first sixteen terms of the A. P.

Let a be the first term and d be the common difference of the A.P we have $a_3+a_7=6$  and $a_3{a}_7=8$

$\Rightarrow \left(a+2d\right)+\left(a+6d\right)=6\text{and}\left(a+2d\right)\times \left(a+6d\right)=8$

$\Rightarrow 2a+8d=6\text{and }\left(a+2d\right)\left(a+6d\right)=8\Rightarrow d=\pm \frac{1}{2}$

Case-I: when $d=\frac{1}{2};a=1$

Case-II: when $d=-\frac{1}{2};a=5\Rightarrow S_{16}=20$