The difference between the squares of two consecutive numbers is 31. Find the numbers.

Let the two consecutive numbers be x, (x + 1).

According to the question, the difference between the squares of two consecutive numbers is 31.

⇒ (x + 1)² - x² = 31

Using the formula,

[∵ (x + y)² = x² + y² + 2xy]

⇒ x² + 1² + 2 $\times$ x $\times$ 1 - x² = 31

⇒ x² + 1 + 2x - x² = 31

⇒ 1 + 2x = 31

⇒ 2x = 31 - 1

⇒ 2x = 30

⇒ x = $\dfrac{30}{2}$

⇒ x = 15

Other number = (x + 1)

= 15 + 1

= 16

Hence, the two consecutive numbers are 15 and 16.