Question

If the infinite A.G.P. $a,-4,-3,4, \ldots \ldots . .$. 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

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