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:
What is the formula to calculate this sum:
The answer is
Here is a calculator that calculates this function for you:
We shall give three different proofs for this formula.
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.
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:
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:
and this leads to: