Prove the following by using the principle of mathematical induction for all `n in N`:`a+a r+a r^2+...+a r^(n-1)=(a(r^n-1))/(r-1)`

Explanation

To prove the statement \( S(n) : a + ar + ar^2 + \ldots + ar^{n-1} = \frac{a(r^n - 1)}{r - 1} \) for all \( n \in \mathbb{N} \) using the principle of mathematical induction, we follow these steps:

### Step 1: Base Case
We first check the base case when \( n = 1 \).

**Left Hand Side (LHS)**:
\[
S(1) = a
...