Take four consecutive natural numbers, what is the relation between the products of the first and the last, and the product of the middle two?

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Solution

Let the four consecutive natural numbers be $x, (x + 1), (x + 2), (x + 3)$ .

Product of first and last $= x (x + 3) = x^{2} + 3 x$
Product of middle two terms $= (x + 1) (x + 2) = x (x + 2) + 1 (x + 2)$
$= x^{2} + 2 x + x + 2$
$= x^{2} + 3 x + 2$

Clearly, we can see that the difference between the product of middle two terms and the product of first and last term is equal to 2.

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