Question

The sum of the squares of two consecutive odd natural numbers is 650. Find the two numbers.

Answer 1

Let the two consecutive od number be 2n + 1 and 2n + 3, where n is a natural number.
Given, (2n + 3)² + (2n + 1)² = 650
⇒ (4n² + 9 + 12n) + (4n² + 1 + 4n) = 650
⇒ 8n² + 10 + 16n = 650
⇒ 8n² + 16n - 640 = 0
⇒ n² + 2n - 80 = 0
⇒  n² + 10n - 8n - 80 = 0
⇒  n(n + 10) - 8(n + 10) = 0
⇒  (n - 8) (n + 10) = 0
Thus, n = 8 as n = -10 is not possible.
Thus, the two consecutive od natural numbers are 17 and 19.