Question

The sum of the squares of three consecutive natural numbers is 110. Determine the numbers.

Answer 1

Let three consecutive natural numbers be x, x + 1 and x + 2.
Then according to problem
(x)² = (x + 1)² + (x + 2)² = 110
⇒ x² + x² + 1 + 2x + x² + 4 + 4x - 110 = 0
⇒ 3x² + 6x - 105 = 0
⇒ x² + 2x - 35 = 0
⇒ x² + 7x - 5x - 35 = 0
⇒ x(x + 7) - 5(x + 7) = 0
⇒ (x + 7) (x - 5) = 0
⇒ x + 7 = 0 or x - 5 = 0
⇒ x = -7 or x = 5
But x = -7 is rejected as it is not a natural number.
Then x = 5
Hence, required numbers are 5, (5 + 1), (5 + 2) i.e., 5, 6 and 7.