# Proving by Mathematical Induction

Prove the following by mathematical induction

Let {$${s_{n}}$$} be the sequence defined by $$s_{0}=\frac{\pi}{4}$$ and $$\forall{n}\geq1$$, $$s_{n}=s_{n-1}+\pi$$.

Show: $$\forall{n}\geq0$$, $$s_{n}=\frac{4n+1}{4}\pi$$.

I think I should first show that $$\forall{n}\geq1$$, $$s_{n}=s_{n-1}+\pi$$ is just the same as $$\forall{n}\geq0$$, $$\frac{4n+1}{4}\pi$$, before I can proceed in proving $$\forall{n}\geq0$$,$$s_{n}=\frac{4n+1}{4}\pi$$ by mathematical induction. However, I had a hard time in showing that the two are just equivalent and with that, I can't start proving the latter.

• $\forall{n}\geq0$, $\frac{4n+1}{4}\pi$ is not a complete mathematical sentence, like "Every day the temperature" is not a complete English sentence. Sep 18, 2019 at 17:48

Base case: $$s_0=\frac{\pi}{4}=\frac{4\times0+1}{4}\pi$$.
Induction step: Suppose the statement is true for $$n$$. Then $$s_{n+1}=s_n+\pi=\frac{4n+1}{4}\pi+\pi=\frac{4n+5}{4}\pi=\frac{4(n+1)+1}{4}\pi.$$
• The value of $s_0$ is already the base case.