Question

Consider the series $r^2+r^4+r^6+r^8+r^{10}+\dots \dots \dots \infty$. The third term is $16$ times the fifth term . The sum of the series is:

Answer 1

Third term$=r^6$

Fifth term$=r^{10}$

Now $r^6=16r^{10}$

$1=16r^4$

$\frac{1}{16}=r^4$

$r=\frac{1}{2}$

$Sum=\frac{a}{1-r}$ (Formula)

$Sum=\frac{r^2}{1-r^2}$

$Sum=\frac{\frac{1}{4}}{1-\frac{1}{4}}$

$Sum=\frac{\frac{1}{4}}{\frac{3}{4}}$

$Sum=\frac{1}{3}$