Sum of Consecutive Odd Numbers

If one sums up a series of consecutive odd numbers starting from one, the total sum is a perfect square of the number of odd numbers added.In other words, the sum of the first $n$ odd numbers has to be $n^2$. Here is what I mean,

$n = 1 \Rightarrow 1 = 1^2 = 1$

$n = 2 \Rightarrow 1 + 3 = 2^2 = 4$

$n = 3 \Rightarrow 1 + 3 + 5 = 3^2 = 9$

$n = 4 \Rightarrow 1 + 3 + 5 + 7 = 4^2 = 16$

$n = 5 \Rightarrow 1 + 3 + 5 + 7 + 9 = 5^2 = 25$

and so on…

Why does this happen? Let’s conceptualize it graphically. Let’s say you have a square of side 1 unit:

How many squares do you need to add to turn this into a 2×2 square?

You need 3 more units!

Next, how many do you need for a 3×3 square?

You need 5 more units!

So in order to turn a square of area 1 sq. units into a square with an area of 2 sq. units, you need to add 3 units. To turn it into a square with an area of 3 sq. units, you need to add 5 more units. How much additional units would a 4×4 square require? 7 more units! 1 , 3 , 5 , 7 , … you get the idea!

Note: A rigorous analytical proof can be found in the book The Higher Arithmetic by R. Davenport