Induktionsbeweis auf Englisch schreiben (Base Case, Inductive Case)

Question

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 :)

Answer 1

Ich finde es ganz prima !

Comments

-- Merci :) --