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Problem 52: (Induction With The Sum Of The First $n$ Natural Numbers)

Use the principle of mathematical induction to prove that for every $n\in \mathbb{N}$, we have $1+2+\cdots+n = \frac{n(n+1)}{2}$.

Standard Induction Hint (This is the same hint for all induction problems):

Let $S$ be the set natural number for which the statement $1+2+\cdots+n = \frac{n(n+1)}{2}$ is true. We want to show that $S=\mathbb{N}$.

  1. First show that the statement is true if you let $n=1$. (Show $1\in S$.)
  2. Then assume that the statement is true if you let $n=k$ for some $k\in \mathbb{N}$.
  3. Use this assumption to then prove that the statement is true when you let $n=k+1$. (This shows if $k\in S$, then $k+1\in S$.)

You can then apply the principle of mathematical induction.

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