1331.14 – Difference of Squares

Find the difference between ${14^2}$ and ${13^2}$. What is the difference between ${181^2}$ and ${180^2}$? Write a formula for the difference of the squares of two consecutive integers, in terms of the smaller of the two.

(Use $n+1$ and $n$.)

Solution

$$14^2 - 13^2 = 27$$

$$181^2 - 180^2 = 361$$

The general pattern is:

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

The figure illustrates this with $n = 4$.