# 3 consecutive numbers

I was playing with some numbers and just realized that:

For any 3 consecutive numbers X, Y and Z: $Y^2$ = (X*Z) + 1

For eg: Consider numbers 171, 172 and 173

$172^2$ = 29584

and

171*173 = 29583

Can anyone tell me if there is any proof for this and what it is known as?

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For your next trick, see what kind of number you get when you multiply four consecutive whole numbers, and add 1. –  Gerry Myerson Jul 4 '12 at 10:22

## 1 Answer

Let the middle number be $x$; the other two are $x-1$ and $x+1$. Basic algebra tells us that $(x-1)(x+1)=x^2-1$, and therefore $x^2=(x-1)(x+1)+1$. (This is true even if $x$ is not an integer.)

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