a + ar + ar
Question:

$a+a r+a r^{2}+\ldots+a r^{n-1}=a\left(\frac{r^{n}-1}{r-1}\right), r \neq 1$

Solution:

Let P(n) be the given statement.

Now,

Step 1;

$P(n)=a+a r+a r^{2}+\ldots+a r^{n-1}=a\left(\frac{r^{n}-1}{r-1}\right), r \neq 1$

$P(1)=a=a\left(\frac{r^{1}-1}{r-1}\right)$

Hence, $P(1)$ is true.

Step 2 :

Suppose $P(m)$ is true.

Then,

$a+a r+a r^{2}+\ldots+a r^{m-1}=a\left(\frac{r^{m}-1}{r-1}\right), r \neq 1$

To show: $P(m+1)$ is true whenever $P(m)$ is true.

That is,

$a+a r+a r^{2}+\ldots+a r^{m}=a\left(\frac{r^{m+1}-1}{r-1}\right), r \neq 1$

$W e$ know that $P(m)$ is true.

Thus, we have :

$a+a r+a r^{2}+\ldots+a r^{m-1}=a\left(\frac{r^{m}-1}{r-1}\right)$

$\Rightarrow a+a r+a r^{2}+\ldots+a r^{m-1}+a r^{m}=a\left(\frac{r^{m}-1}{r-1}\right)+a r^{m} \quad$ [Adding $a r^{m}$ to both sides]

$\Rightarrow P(m+1)=a\left(\frac{r^{m}-1+r \cdot r^{m}-r^{m}}{r-1}\right)$

$\Rightarrow P(m+1)=a\left(\frac{r^{m+1}-1}{r-1}\right), r \neq 1$

Thus, $P(m+1)$ is true.

By the principle of $m$ athematical $i$ nduction, $P(n)$ is true for all $n \in N$.