The Sum of the first n natural numbers solution

[showmyiq]

Because we are going to use that very often, I would like to write down simple and beautiful proof.

We need to calculate the sum of all natural numbers from 1 to n.

\begin{equation}
S_n = 1 + 2 + 3 + .... + n
\end{equation}

Let’s arrange it like that:

Sn = 1 + 2 + 3 + … + (n-1) + n

If we multiply the equation both sides by 2, we get this:

[1 + 2 + 3 + … + (n-1) + n] + [1 + 2 + 3 + … + (n-1) + n] = 2*Sn

Let’s rearrange the left side like that

[1 + 2 + 3 + … + (n-1) + n] +
[n + (n-1) + ….+ 2 +1] =
=2*Sn

Do you see the pattern now? If we add the first parts of the elements in the left side of the equation, we get n+1. If we add the second parts – 2 and (n-1), we again get n+1. If we continue like that till the end elements, we again receive n+1. We have a total of n pairs, so the result of the left side of the equation will be n pairs multiplied by (n+1), which is their value. So we get

n*(n+1) = 2*Sn

So at the end we receive:

\begin{equation}
S_n =\frac {n*(n+1)} {2}
\end{equation}

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