Prove the following Theorem using PMI Theorem on the Geometr

Prove the following Theorem using PMI

Theorem on the Geometric Series: 1 + a + a2 + a3 + ...+ an-1 = (an - 1) / (a -1) for all natural numbers.

Solution

For the base case, n = 1, we have that 1 = (a^1 - 1) / (a - 1) = (a - 1) / (a - 1) as required.

For the inductive step, assume that 1 + a + a^2 + a^3 + ... + a^(n-1) = (a^n - 1) / (a - 1). Then consider

1 + a + a^2 + a^3 + ... + a^(n-1) + a^n = (a^n - 1) / (a - 1) + a^n (by assumption)

1 + a + a^2 + a^3 + ... + a^(n-1) + a^n = (a^n - 1) / (a - 1) + a^n (a - 1)(a - 1)

1 + a + a^2 + a^3 + ... + a^(n-1) + a^n = [a^n - 1 + a^n(a - 1)] / (a - 1)

1 + a + a^2 + a^3 + ... + a^(n-1) + a^n = [a^n - 1 + a^(n+1) - a^n] / (a - 1)

1 + a + a^2 + a^3 + ... + a^(n-1) + a^n = (a^(n+1) - 1) / (a - 1) + a^n

as was to be shown.