Question

Find a formula for $S_n$, the sum of the first $n$ terms of the geometric sequence.

Answer 1

Find the sum of first $n$ terms of the geometric sequence.

Consider the G.P, $a,ar,a{r}^2,a{r}^3,......a{r}^{n-1}$

Let the sum of n terms be $S_n$, first term be $a$ and the common ratio of G.P. be $r$.

Then,$S_n=a+ar+a{r}^2......a{r}^{n-1}$ …...…….$\left(1\right)$

For $r=1$, $S_n=a+a+a+.....a=na$

For $r\ne 1$,

Multiply both sides of $S_n$ by $r$, we get,

$r{S}_n=ar+a{r}^2+a{r}^3......a{r}^n$ ………….$\left(2\right)$

Subtracting $\left(2\right)$ from $\left(1\right)$, we get,

$\Rightarrow r{S}_n-S_n=(ar+a{r}^2+a{r}^3......a{r}^{n-1}+a{r}^n)-(a+ar+a{r}^2+a{r}^3......a{r}^{n-1})$

$\Rightarrow r{S}_n-S_n=a{r}^n-a$

$$\begin{gathered} \Rightarrow (r-1)S_n=a(r^n-1) \\ \Rightarrow S_n=\frac{a(r^n-1)}{(r-1)} \end{gathered}$$

Hence, sum of $n$ terms of the geometric sequence is ${\mathit{S}}_{\mathbf{n}}\mathbf{=}\frac{\mathbf{a}\mathbf{(}{\mathbf{r}}^{\mathbf{n}}\mathbf{}\mathbf{-}\mathbf{}\mathbf{1}\mathbf{)}}{\mathbf{(}\mathbf{r}\mathbf{-}\mathbf{1}\mathbf{)}}$.