The sum of the squares of three consecutive odd numbers is 2531. Find the numbers.

Let the three consecutive odd numbers be x, x + 2, x + 4.

Given,

The sum of the squares of three consecutive odd numbers is 2531.

⇒ x² + (x + 2)² + (x + 4)² = 2531

⇒ x² + x² + (2)² + 2 × 2 × x + x² + (4)² + 2 × 4 × x = 2531

⇒ x² + x² + 4 + 4x + x² + 16 + 8x = 2531

⇒ 3x² + 12x + 20 = 2531

⇒ 3x² + 12x + 20 - 2531 = 0

⇒ 3x² + 12x - 2511 = 0

⇒ 3x² + 93x - 81x - 2511 = 0

⇒ 3x(x + 31) - 81(x + 31) = 0

⇒ (3x - 81)(x + 31) = 0

⇒ (3x - 81) = 0 or (x + 31) = 0     [Using zero-product rule]

⇒ 3x = 81 or x = -31

⇒ x = $\dfrac{81}{3}$ or x = -31

⇒ x = 27 or x = -31:

Case 1: If x = 27,

x + 2 = 27 + 2 = 29

x + 4 = 27 + 4 = 31

Case 2: If x = -31,

x + 2 = -31 + 2 = -29

x + 4 = -31 + 4 = -27

Hence, the three consecutive odd numbers are 27, 29, 31 or -31, -29, -27.