Sommatoria

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Index

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${\sum }_{i=0}^{n}i=\frac{n(n+1)}{2}$ Gauss ha trovato questa formula a 8 anni 🤡

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Dimostrazione per induzione della sommatoria

Dimostrare: ${\sum }_{i=0}^{n}i=\frac{n(n+1)}{2}\forall n\in \mathbb{N}$ Caso Base: $P(0)={\sum }_{i=0}^{0}i=\frac{n(n+1)}{2}$ Ipotesi Induttiva: $$ \begin{align} &P(n)= \sum_{i=0}^{n}i=\frac{{n(n+1)}}{2}\

& P(n+1)=\sum_{i=0}^{n+1}i=\textcolor{orange}{\frac{{(n+1)(n+2)}}{2}}\ \end{align}$\ast \ast Dim:\ast \ast$\begin{align}

& P(n+1)=\sum_{i=0}^{n+1}i=\sum_{i=0}^{n}i+(n+1)= \ &\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ = \frac{{n(n+1)}}{2}+n+1= \frac{{n(n+1)+2(n+1)}}{2}= \ & \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ = \textcolor{orange}{\frac{{(n+1)(n+1)}}{2}}\ \end{align}$$