Question

The sum of an infinite geometric progression is $2$ and the sum of the geometric progression made from the cubes of this infinite series is $24$. Then find the value of $(-2)$ times the common ratio ?

Answer 1

Let the first term of the given G.P is '$a$' and common ratio is '$r$'.
Using given condition, we get
$\frac{a}{1-r}=2$ ........ (1) and $\frac{a^3}{1-r^3}=24$ where $|r|<1$ ....... (2)
Taking cube on both side of equation (1) and substitute $a^3$ in (2), we get
$\frac{(1-r{)}^3}{1-r^3}=\frac{24}{8}$
$⟹3(1+r+r^2)=(1-r{)}^2=1-2r+r^2$
$⟹2r^2+5r+2=0$
$⟹(2r+1)(r+2)=0$
$\therefore r=-\frac{1}{2}=-0.5$ $[\because |r|<1]$

Hence, $-2\times r=-2\times \left(\frac{-1}{2}\right)=1$