The sum of the first n natural numbers

What is the sum of the first n natural numbers?

Let's look at this problem for n=1, 2, 3, 4, and 5 and calculate the sum:

1=1
1+2=3
1+2+3=6
1+2+3+4=10
1+2+3+4+5=15

What is the formula to calculate this sum:  Related sums:
The sum of the first n odd natural numbers
The sum of the squares of the first n natural numbers

Here is a calculator that calculates this function for you:

n:

1+2+...+n:

We shall give three different proofs for this formula.

Proof 1: This is an example for n = 5. We see that we have a big rectangle with the its sides 5 and 5+1. The rectangle has 2(1 + 2 + 3 + 4 + 5) squares inside. So 2(1 + 2+ 3 + 4 + 5) = 5(5+1) and 1 + 2 + 3 + 4 + 5 = The same way, for any n, we construct a rectangle with its sides n and n+1 that has 2(1+2+...+n) squares inside.

Proof 2:  (1+2+....+(n-1)+n) +
(n+(n-1)+....+2+1) = if we look at the sum above we notice that we have n columns of numbers and that on each column we have two numbers with the sum n+1, so:  Proof 3:

Let's write like this: , . It's shorter.

We will use induction to prove that: 1) For n = 1 it is true that 2)We know that is true and we want to prove it for n+1.
We have to prove that: if we add n+1 to each side of the next identity: we have: equivalent with: and this leads to: 