ThmDex – An index of mathematical definitions, results, and conjectures.

Result R5644 on D226: Set of initial natural numbers

Complement formula for the sum of initial natural numbers

Formulation 0

Let $n \in \mathbb{N}$ be a D996: Natural number.

Then \begin{equation} \sum_{n = 0}^N n = \sum_{n = 0}^N (N - n) \end{equation}

Proofs

Proof 0

Let $n \in \mathbb{N}$ be a D996: Natural number.

We have \begin{equation} \begin{split} \sum_{n = 0}^N n & = 0 + 1 + 2 + \cdots + (N - 2) + (N - 1) + N \\ & = N + (N - 1) + (N - 2) + \cdots + 2 + 1 + 0 \\ & = \sum_{n = 0}^N (N - n) \end{split} \end{equation} $\square$