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The sum of the first numbers

The sum $1 + 2 + \dots + n$ is $n(n + 1)/2$. The formula is proved by induction. The base case is $n = 1$, where both sides are 1. In the induction step you assume that the formula holds for $k$, add the next number $k + 1$, and show that you get the formula for $k + 1$. Then it holds for every natural number. You must use the hypothesis in the step.

Symbols

Example

$1 + 2 + \dots + 100 = \dfrac{100\cdot 101}{2} = 5050$. The base case $n = 1$ gives $\dfrac{1\cdot 2}{2} = 1$.

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Part of Mathematics R2: Proof by induction.