Question

Show that the ratio of the sum of first $n$ terms of a G.P. to the sum of terms from $(n+1{)}^{th}$ to $(2n{)}^{th}$ term is $1/r^n$.

Answer 1

Let terms be $r,r^2,r^3.....r^n,r^{n+1}\dots .r^{2n}$

Sum of n terms of gp $=\frac{a(r^n-1)}{r-1}$

$\Rightarrow \frac{r(r^n-1)}{r-1}$ ………$\left(1\right)$

Sum of $(n+1)$ terms of $\left(2n\right)$ terms of g.p

$\Rightarrow \frac{r^{n+1}(r^n-1)}{r-1}$ ………$\left(2\right)$

$\therefore$ Ratio of $\left(1\right)$ by $=\frac{\frac{r(r^n-1)}{r-1}}{\frac{(r^n+1)(r^n-1)}{r-1}}=\frac{r}{r^n+1}$

$\Rightarrow \frac{1}{r^n}$

Hence proved.