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Sum of the first odd numbers

The sum of the first $n$ odd numbers is $n^2$. The first is 1, the second is 3, and the $n$th is $2n - 1$. The base case is $1 = 1^2$. In the step you add the next odd number $2k + 1$, and $k^2 + 2k + 1$ is the square of $k + 1$. A concrete count checks the formula, while induction shows that it holds for every $n$.

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Example

The first 50 odd numbers end at $99$, and the sum is $50^2 = 2500$. The first four are $1 + 3 + 5 + 7 = 16 = 4^2$.

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Part of Mathematics R2: Proof by induction.