Prove by induction that $\sum_{i=1}^{n} 2i=(n+1)n$, for every positive integer n.

Question

Can anyone explain the concept behind this? I just don't get how I should proceed with it? Like each step, why and how is it done?

Prove by induction that $\displaystyle\sum_{i=1}^{n} 2i=(n+1)n$, for every positive integer $n$.

Answer 1

There is a proof in here. The standard version of it is $\displaystyle\sum_{i=1}^{n} i=\dfrac{n(n+1)}{2}$

By induction: First, it is valid for $n=1$. Second, suppose that it is correct for $n=k$, i.e., $\displaystyle\sum_{i=1}^{k} i=\dfrac{k(k+1)}{2}$ and by this assumption prove for $n=k+1$.