Question

Find the n^th terms of the sequences:  0.2, 0.22, 0.222, …

Answer 1

Let t₁ = 0.2, t₂ = 0.22, t₃ = 0.222 and so on.
t₁ = 0.2
t₂ = 0.22 = 0.2 + 0.02 
t₃ = 0.222 = 0.2 + 0.02 + 0.002 
∴ t_n = 0.2 + 0.02 + 0.002 + ... upto n terms
But 0.2, 0.02, 0.002, … upto n terms are in
G.P. with a = 0.2 and r = 0.1
∴ t_n = the sum of first n terms of the G.P.

∴ t_n = `0.2{(1 - (0.1)^"n")/(1 - 0.1)}`

∴ t_n = `0.2/0.9{1 - (0.1)^"n"}`

∴ t_n = `2/9{1 - (0.1)^"n"}`