Question

the sum of first six term is six.the product of second and fifth term is -80.find the terms of an ap

Answer 1

$\text{Let the required A.P be}$

$\text{a, a+d, a+2d,.........}$

$\textbf{Given:}$

$S_6=6$

$\implies\frac{n}{2}[2a+(n-1)d]=6$

$\implies\frac{6}{2}[2a+5d]=6$

$\implies\,2a+5d=2$

$\implies\,a=\frac{2-5d}{2}$ ....(1)

$\text{Also, }\;t_2{\times}t_5=-80$

$\implies(a+d)(a+4d)=-80$

$\implies(\frac{2-3d}{2})(\frac{2+3d}{2})=-80$

$\implies\frac{2^2-9d^2}{4}=-80$

$\implies\,4-9d^2=-320$

$\implies\,-9d^2=-324$

$\implies\,d^2=36$

$\implies\bf\,d=\pm\,6$

$\text{When d=6, a=-14}$

$\textbf{The A.P is -14, -8, -2.........}$

$\text{When d=-6, a=16}$

$\textbf{The A.P is 16, 10, 4.........}$