Physics and Maths Tutoring from Michael A Smith.

Take two consecutive square number, say $16$ and $25$ , their difference is $25 - 16 = 9$ which is odd. So we have some evidence to support our claim that the difference between two consecutive square numbers if odd. However, this method is not going to be much use as there are infinity many pairs of consecutive square number, so we have to find a way that covers all pairs.

Warm-up:

Perfect Squares

Expanding Brackets

The Problem:

Show that the difference of two conscutive perfect squares is an odd number using algebra.

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Solution:

Let $n$ be an integer (or $x \in ℤ$ )
then $n^{2}$ and ${(n + 1)}^{2}$ are conscutive perfect squares. The difference is given by

{(n + 1)}^{2} - n^{2}

Let's simplify this expression be expanding the brackets, $n^{2} + 2 n + 1 - n^{2} = 2 n + 1$

Now $2 n$ is an even number. Adding one an even number gives an odd, so $2 n + 1$ is odd for all $n$ . Proving that the difference of two consecutive perfect squares is an odd integer.

Proof that the Difference of Two Consecutive Perfect Squares is Odd

Warm-up:

The Problem:

Solution:

Further Reading: