Prove, by induction, the formula for the sum of the first n terms of a geometric series - Leaving Cert Mathematics - Question 4 - 2012

We start by defining the series:

Let P(n):

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

Base Case (n = 1): Check P(1):

$S_1 = a = \frac{a(1-r^1)}{1-r} = a$

This is true.

Inductive Step: Assume P(k) holds, meaning:

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

Then, we must show P(k + 1):

$S_{k+1} = S_k + ar^{k}$

Substituting the assumption:

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

Combine the terms:

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

This completes the inductive step, proving that P(n) holds for all n ∈ ℕ.