Q.Prove the following by using the principle of mathematical induction for all $n \in N$: $a + ar + ar^2 + \ldots + ar^{n-1} = \dfrac{a(r^n - 1)}{r - 1}$

Let $P(n)$ be the statement (for fixed $a\ne0$, $r\ne1$)

Base case: For $n=1$, LHS $=a$, and

So $P(1)$ is true.

Inductive step: Assume $P(k)$ is true for some $k\ge1$:

We must show $a+ar+\ldots+ar^{k-1}+ar^k=\dfrac{a(r^{k+1}-1)}{r-1}$.

Adding the next term $ar^k$ to both sides of the hypothesis: …

Geometric-series induction. Add the next term $ar^k$ to the closed form, combine over the common denominator $(r-1)$, and cancel the $-r …

• Assuming $r=1$ implicitly (the formula and this proof require $r\ne1$, since the derivation divides by $r-1$; the $r=1$ case is trivially $na$ and must be excluded or handled separately). …