Question

Find the n^th terms of the sequences: 0.5, 0.55, 0.555, …

Answer 1

Let t₁ = 0.5, t₂ = 0.55, t₃ = 0.555 and so on.
t₁ = 0.5
t₂ = 0.55 = 0.5 + 0.05
t₃ = 0.555 = 0.5 + 0.05 + 0.005
∴ t_n = 0.5 + 0.05 + 0.005 + ... upto n terms
But 0.5, 0.05, 0.005, … upto n terms are in
G.P. with a = 0.5 and r = 0.1
∴ t_n = the sum of first n terms of the G.P.

∴ t_n = `0.5{(1 - (0.1)^"n")/(1 - 0.1)}`

∴ t_n = `0.5/0.9{1 - (0.1)^"n"}`

∴ t_n = `5/9{1 - (0.1)^"n"}`