Let P(n): a + ar + ar² + … + ar^{n - 1} =

For n =1

a = a

a = a

Since, P(n) is true for n = 1

Let P(n) is true for n = k , so

P(k): a + ar + ar² + … + ar^{k - 1} = - - - - - - - (1)

We have to show that,

a + ar + ar² + … + ar^{k - 1} + ar^k =

Now,

{ a + ar + ar² + … + ar^{k - 1}} + ar^k

= using equation (1)

=

=

=

Therefore, P(n) is true for n = k + 1

Hence, P(n) is true for all n ∈ N