If the second term of a G.P is 2 and its sum to infinity is 8, find its fifth term.

For a G.P., the second term is ar = 2 and the sum to infinity is a/(1 – r) = 8.

From ar = 2, a = 2/r. Put this in: 2/(r(1 – r)) = 8, so r(1 – r) = 1/4. This gives r² – r + 1/4 = 0, so (r – 1/2)² = 0 and r = 1/2.

Then a = 2 ÷ 1/2 = 4. The fifth term is ar⁴ = 4 × (1/2)⁴ = 4/16 = 1/4.