Question

The 5 terms of an AP are a1, a2, a3, and a5. Given:a1+a3+a5=-12 and a1. a2. a3=8.
Find the first term and the common difference.

Answer 1

Question : The 5 terms of an A.P. are a₁, a₂, a₃, a₄ and a₅. Given : a₁ + a₃ + a₅ = - 12 and a₁.a₂.a₃ = 8. Find the first term and the common difference.

Solution :

Let, the first term = a

and the common difference = d

Then,

• a₁ = a
• a₂ = a + d
• a₃ = a + 2d
• a₄ = a + 3d
• a₅ = a + 4d

Given : a₁ + a₃ + a₅ = - 12

⇒ a + a + 2d + a + 4d = - 12

⇒ 3a + 6d = - 12

⇒ a + 2d = - 4 .....(1)

and a₁.a₂.a₃ = 8

⇒ a (a + d) (a + 2d) = 8

⇒ a (a + d) (- 4) = 8 , by (1)

⇒ a (a + d) = - 2

⇒ (- 4 - 2d) (- 4 - 2d + d) = - 2

⇒ (- 4 - 2d) (- 4 - d) = - 2

⇒ (d + 4) (2d + 4) = - 2

⇒ 2d² + 12d + 16 + 2 = 0

⇒ 2d² + 12d + 18 = 0

⇒ d² + 6d + 9 = 0

⇒ (d + 3)² = 0

⇒ d + 3 = 0

∴ d = - 3

From (1), we get

a + 2 (- 3) = - 4

⇒ a - 6 = - 4

⇒ a = 6 - 4

∴ a = 2

Thus, the first term of the A.P. is 2 and the common difference is (- 3).