Question

An infinite geometric progression a₁ , a₂ , a₃ ,... has the property that a_n = 3(a_{n+ l} + 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 1

Correct Answer - Option 3 : 3/32

Calculation:

For any n ≥ 1, a_n = 3 (a_n+1 + a_n+1 + ........)
∴   a₁ = 3 (a2 + a₃ + -----) or r = 1⁄4 and
a₁ + a₂ + ---- = (4 × a₁)/3 = 32 (as given)

∴ a₁ = 24
The GP is 24, 6, 1.5, 1.5/4, 1.5/16

⇒ a₅ = 3/32

For an infinite GP if the common-ratio is less than 1 the sum of series = a/(1 - r)