Mich würde interessieren, ob ich einen Induktionsbeweis wie folgt formal korrekt auf Englisch schreiben kann:
Karl Friedrich Gauss - a famous mathematician - invented a formula with which one can calculate the sum of the first \(n\) natural numbers without summing up each element one by one. $$ \sum\limits_{i=1}^{n}{i}=\frac{n\cdot (n+1)}{2} $$ We're going to prove this formula for all \(n\in\mathbb{N}\) by mathematical induction:
Base Case Let \(n = 1\) be the first value for which $$ \sum\limits_{k=1}^{n}{i}=\frac{n\cdot (n+1)}{2} $$ is true. We evaluate the left side and the right side of the given formula for \(n=1\):
- \(\sum\limits_{k=1}^{n}{i}=\sum\limits_{i=1}^{1}{k}=1\)
- \(\frac{n\cdot (n+1)}{2}=\frac{1\cdot (1+1)}{2}=\frac{2}{2}=1\)
Both sides are equal and the given formula is true for \(n=1\) \(\checkmark\).
Inductive Step Let \(k\in\mathbb{N}\) be given and suppose the formula is true for \(n = k\). If we want to prove that the statement is true for \(n = k+1\), then $$ \begin{array}{lcl} \sum\limits_{i=1}^{k+1}{i}&=&\sum\limits_{i=1}^{k}{i} + (k+1)\\ &=&\underbrace{\frac{k\cdot (k+1)}{2}}_{\text{by induction hypothesis}} +(k+1)\\ &=&\frac{k\cdot(k+1)+2(k+1)}{2}\\ &=&\frac{(k+1)\cdot (k+2)}{2}\\ &=&\frac{(k+1)\cdot ((k+1)+1)}{2}\\ \end{array} $$Thus, the formula holds for \(n = k + 1\), and the proof of the induction step is complete.
By the principle of induction, the formula is true for all \(n\in\mathbb{N}\). \(\square\)
Thanks in advance :)
