The sum of three numbers in A.P. is 15 and the sum of the squares of the extreme terms is 58. Find the numbers.

Let the numbers be a - d, a, a + d.

According to question,

⇒ a - d + a + a + d = 15

⇒ 3a = 15

⇒ a = 5.

Given,

Sum of the squares of the extreme terms is 58.

⇒ (a - d)² + (a + d)² = 58

⇒ (5 - d)² + (5 + d)² = 58

⇒ 25 + d² - 10d + 25 + d² + 10d = 58

⇒ 50 + 2d² = 58

⇒ 2d² = 8

⇒ d² = 4

⇒ d = ±2

Let d = 2,

Numbers = (5 - 2), 5, (5 + 2) = 3, 5, 7.

Let d = -2,

Numbers = (5 - (-2)), 5, (5 + (-2)) = 7, 5, 3.

Hence, numbers = 3, 5, 7 or 7, 5, 3.