If the sum of an infinite G.P.

Question

If the sum of an infinite G.P. $a, ar, ar^2, ar^3,\ldots$ is $15$, and the sum of the squares of its terms is $150$, then the sum of $ar^2, ar^4, ar^6,\ldots$ is:

Accepted Answer

$\dfrac{a}{1-r}=15$; $\dfrac{a^2}{1-r^2}=150 \Rightarrow \dfrac{a}{1-r}\cdot\dfrac{a}{1+r}=150 \Rightarrow \dfrac{a}{1+r}=10.$

Dividing: $\dfrac{1-r}{1+r}=\dfrac{10}{15}=\dfrac23 \Rightarrow r=\dfrac15$, and $a=12$.

$$\frac{ar^2}{1-r^2} = \frac{12\times\frac{1}{25}}{\frac{24}{25}} = \frac{12}{24} = \frac12.$$

Solutions · 1