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

 

[CellCoree]

The reason why the sum of odd numbers always results in a square number is due to the mathematical concept of perfect squares and odd numbers. A perfect square is a number that can be represented as the product of two equal factors, such as 4 = 2 x 2 or 9 = 3 x 3. Odd numbers, on the other hand, are numbers that cannot be divided evenly by 2.

When we add consecutive odd numbers together, we are essentially adding two consecutive integers. For example, 1+3 = 4, which is equal to 2 x 2. Similarly, 1+3+5 = 9, which is equal to 3 x 3. This pattern continues as we add more odd numbers, resulting in the sum always being a perfect square.

In general, the sum of n consecutive odd numbers can be represented as n^2, which is a perfect square. This can be proven through algebraic manipulation, but the concept remains the same.

Therefore, the reason why the sum of odd numbers always results in a square number is due to the relationship between odd numbers and perfect squares. This is a fundamental concept in mathematics and has been observed and studied by mathematicians for centuries.