An infinite geometric progression a1, a2, a3, ... has the property

• Progressions

An infinite geometric progression a₁, a₂, a₃, ... has the property that a_n = 3(a_n+1 + a_n+2 +…) for every n ≥ 1. If the sum a₁ + a₂ + a₃ + … = 32, then a₅ is

1. 1/32
2. 2/32
3. 3/32
4. 4/32

Answer

For any n ≥ 1, a_n = 3(a_n+1 + a_n+2 + …)

a₁ = 3(a₂ + a₃ + …) or r = 1/4

a₁ + a₂ + a₃ + ... = 4a₁/3 = 32 (given)

a₁ = 24

The GP is 24, 6, 1.5, 1.5/4, and so on.

a₅ = 1.5/16 = 3/32

The correct option is C.