The sum of the first n odd numbers

Exercise

Show that for every positive integer ,

Try to find both an algebraic proof and a geometric explanation before reading on.

Proof by induction

For , the claim is , which is true.

Suppose the claim holds for some . Adding the next odd number gives

This proves the claim by induction.

A geometric explanation

Start with one square. To enlarge an square to an square, add a row of squares, a column of squares, and one corner square: new squares altogether.

Each successive odd number supplies exactly the next layer.

Further question

What is the sum of the odd numbers from to , where ?

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