Find three numbers in A.P., whose sum is 15 and the product is 80.

Let the three numbers in the Arithmetic Progression (A.P.) be a - d, a, a + d.

Given,

The sum of the three numbers is 15.

⇒ a - d + a + a + d = 15

⇒ 3a = 15

⇒ a = $\dfrac{15}{3}$

⇒ a = 5.

The middle term of the A.P. is 5.

Given,

The product of the three numbers is 80.

⇒ (a - d)(a)(a + d) = 80

⇒ (5 - d)(5)(5 + d) = 80

⇒ (5 - d)(5 + d) = $\dfrac{80}{5}$

⇒ (5 - d)(5 + d) = 16

⇒ 5² - d² = 16

⇒ 25 - d² = 16

⇒ d² = 25 - 16

⇒ d² = 9

⇒ d = $\sqrt{9}$

⇒ d = 3 or d = -3.

Case 1 : If a = 5 and d = 3

⇒ a - d = 5 - 3 = 2

⇒ a = 5

⇒ a + d = 5 + 3 = 8.

Case 2 : If a = 5 and d = -3

⇒ a - d = 5 - (-3) = 8

⇒ a = 5

⇒ a + d = 5 + (-3) = 2.

Hence, the numbers are 2, 5, 8.