Question

If the infinite A.G.P. $a,-4,-3,4,\dots \dots ..$. has a finite sum, then find the value of $(a+d)$ where $d$ is common difference of corresponding A.P. associated to A.G.P.

Answer 1

Answer: 8

Let $S=a+(a+d)r+(a+2d)r^2$
$a+(a+d)r=-4,(a+2d)r^2=-3,(a+3d)r^3=4$
$\Rightarrow a+d=-\frac{4}{r},a+2d=-\frac{3}{r^2},a+3d=\frac{4}{r^3}$
$\Rightarrow d=-\frac{3}{r^2}+\frac{4}{r},d=\frac{4}{r^3}+\frac{3}{r^2}$
$\therefore 2r^2-3r-2=0\Rightarrow r=2,-\frac{1}{2}$
For infinite A.G.P. $|r|<1\Rightarrow r=-\frac{1}{2}$
$d=-\frac{3}{r^2}+\frac{4}{r}=-12-8=-20$
$a+d=\frac{4}{r}=8.$