a1, a2, a3, a4, a5 are the first five terms of an A.P. such that a1 +

1.	a1, a2, a3, a4, a5 are the first five terms of an A.P. such that a1 + a3 + a5 = –12 and a1.a2.a3 = 8. Find the first term and common difference.
Answer» Let a₁ = a₃ – 2d, a₂ = a₃ – d, a₃ = a₃, a₄ = a₃ + d, a₅ = a₃ + 2d Then a₁ + a₃ + a₅ = –12, (given) ⇒ a₃ – 2d + a₃ + a₃ + 2d = –12 ⇒ 3a₃ = –12 ⇒ a₃ = – 4 Also, a₁ . a₂ . a₃ = 8 (given) ⇒ a₁ . a₂ = –2 (∵ a₃ = – 4) ⇒ (a₃ – 2d) (a₃ – d) = –2 ⇒ (– 4 – 2d) (– 4 – d) = – 2 ⇒ (2 + d) (4 + d) = 1 ⇒ d² + 6d + 9 = 0 ⇒ (d + 3)² = 0 ⇒ d = – 3 ∴ a₁ = a₃ – 2d = – 4 –2 (–3) = 2.

a1, a2, a3, a4, a5 are the first five terms of an A.P. such that a1 + a3 + a5 = –12 and a1.a2.a3 = 8. Find the first term and common difference.

Answer»

Let a₁ = a₃ – 2d, a₂ = a₃ – d, a₃ = a₃, a₄ = a₃ + d, a₅ = a₃ + 2d

Then a₁ + a₃ + a₅ = –12, (given)

⇒ a₃ – 2d + a₃ + a₃ + 2d = –12

⇒ 3a₃ = –12

⇒ a₃ = – 4

Also, a₁ . a₂ . a₃ = 8 (given)

⇒ a₁ . a₂ = –2 (∵ a₃ = – 4)

⇒ (a₃ – 2d) (a₃ – d) = –2

⇒ (– 4 – 2d) (– 4 – d) = – 2

⇒ (2 + d) (4 + d) = 1

⇒ d² + 6d + 9 = 0 ⇒ (d + 3)² = 0 ⇒ d = – 3

∴ a₁ = a₃ – 2d = – 4 –2 (–3) = 2.