Proof That The Sum The First N Odd Integers is a Square

The sum of the first n odd integers is a square. This is a theorem and can easily be proven if you have already learned proof by mathematical induction. Even though it sounds like a boring theorem, it is actually very interesting if represented visually. Consider the following diagrams and their numerical representations.

1+3=4

1 + 3 = 2^2

 1+3+5=9

1 + 3 + 5 = 3^2

Can you see the pattern? Let’s have some more examples. 

1+3+5+7=16

1 + 3 + 5 + 7 = 4^2

 first n odd integers

1 + 3 + 5 + 7 + 9 = 5^2

For teachers, from here, students who are not familiar with this theorem can make a conjecture that the sum of the first n integers is a square number or

1 + 3 + 5 + \cdots + (2n - 1) = n^2

where n = 1, 2, 3, \dots.

In addition, it is also possible to ask the students to explain in their own words why the theorem is true if proof by mathematical induction has not beed discussed yet.

The illustration above is an example of proof without words, a visual representation of mathematical statements that helps a reader see why a particular statement is true or false.

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