Question

Prove by Induction $a+ar+{ar}^2+.....$ up to n terms =$\frac{a(r^n-1)}{r-1},r\ne 1$

Answer 1

consider the base case $n=1$

$LHS=a,RHS=a\left(\frac{r^1-1}{r-1}\right)=a$

Hence, statement holds for $n=1$

Now assume statement is valid for $n=k$

i.e

$a+ar+a{r}^2.......a{r}^{k-1}=a\left(\frac{r^k-1}{r-1}\right)$

Adding $a{r}^k$ to both sides, we get

$a+ar+a{r}^2........a{r}^k=a(\frac{r^k-1}{r-1}+\frac{r^{k+1}-r^k}{r-1})$

$a+ar+a{r}^2........a{r}^k=a\left(\frac{r^{k+1}-1}{r-1}\right)$

i.e statement holds for $n=k+1$ also

Hence, by Mathematical function,

Statement is true for all $nϵN$