Numbers are 2,5 and 8 or -3, 5 and 13.
Explanation:
- Step 1: Given the sum of three numbers in AP is 15 and their product is 80.
Let the numbers be a, a+d and a+2d.
- Step 2: Form equations with the given details.
⇒ a + a + d + a + 2d = 15
⇒ 3a + 3d = 15
⇒ a + d = 5 ---- (1)
⇒ d = 5 - a ---- (2)
Also, a(a + d)(a + 2d) = 80 ----- (3)
- Step 3: Substitute (1) and (2) in (3)
⇒ a × 5 ×(a + 2d) = 80
⇒ a (a + 2d) = 16
⇒ a² + 2a (5-a) = 16
⇒ a² + 10a - 2a² = 16
⇒ a² - 10a + 16 = 0
⇒ a² - 2a - 8a + 16 = 0
⇒ a(a - 2) - 8(a - 2) = 0
⇒ (a - 8)(a - 2) = 0
⇒ a = 2, 8
- Step 4: Find d by substituting a in eq(2)
⇒ d = 5 - a = 3 (when a = 2)
or d = 5 - a = -3 (when a = 8)
⇒ a = 2 and d = 3 or a = - 3 and d = 8
Numbers are 2, 5 and 8 or -3, 5 and 13