Question

If $a,b,c$ be the respective sums of the $n$ terms, next $n$ terms and of G.P. then prove that $a,b,c$ are in G.P.

Answer 1

Let x be the first term & r be the common ratio of the G.P

$a=\frac{x(1-r^n)}{1-r}$ ………$\left(1\right)$

$b=\frac{x(1-r^{2n})}{1-r}-a$

$\Rightarrow a+b=\frac{x(1-r^{2n})}{1-r}$ …..$\left(2\right)$

$c=\frac{x(1-r^{3n})}{1-r}-(a+b)$

$\Rightarrow a+b+c=\frac{x(1-r^{3n})}{1-r}$ ……$\left(3\right)$

$\left(2\right)÷\left(1\right)$

$\frac{a+b}{a}=\frac{1-r^{2n}}{1-r^n}$

$1+\frac{b}{a}=1+r^n$

$b=a{r}^n$

$\left(3\right)÷\left(2\right)$

$\frac{a+b+c}{a+b}=\frac{1-r^{3n}}{1-r^{2n}}$

$1+\frac{c}{a+b}=\frac{1+r^n+r^{2n}}{1+r^n}$

$\frac{c}{a+b}=\frac{r^{2n}}{1+r^n}$

$\frac{c}{a+a{r}^n}=\frac{r^{2n}}{1+r^n}$

$\frac{c}{a(1+r^n)}=\frac{r^{2n}}{1+r^n}$

$c=a{r}^{2n}$

We can say that $a,a{r}^n,a{r}^{2n}$ are in G.P.

$\therefore$ a, b, c are in G.P.