Why is the sum of odd numbers is a square

[madah12]

ok so I was messing with the calculator by adding only odd numbers together
1= 1^2 ,1+3=2^2 , 1+3+5=3^3 , 1+3+5+7=4^2,1+3+5+7+9=5^2 I continued this till 27 and i always got a square why is that?

 

[fedaykin]

I love finding patterns like that.

You may be interested in seeing this:

http://mathforum.org/library/drmath/view/56866.html

It explains a little bit about induction, and provides a proof by induction.

 

[stevenb]

It's not hard to show this algebraically, but geometrically is easier to visualize.

Make a big square out of 4 smaller squares. Now, how
many small squares need to be placed on two adjacent sides to make a bigger square? You need 2 on one side and 2 on the other, and then you need one more square to fill the missing gap. For any big square made of N^2 small squares, you need 2N+1 additional small squares to make the next larger square. Well, 2N+1 is always the next larger odd integer in the sequence.

 

[AlephZero]

Draw a small sqaure.
Add 3 squares in an L shape round the edge, to make a 2x2 square
Add 5 more squares in an L shape to make a 3x3 square
Repeat till you get bored