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 odd numbers has to be . Here is what I mean,

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}

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