Question

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 1

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.