Question

Find the sum to n terms: 0.7 + 0.77 + 0.777 + ...

Answer 1

S_n = 0.7 + 0.77 + 0.777 + … upto n terms
= 7(0.1 + 0.11 + 0.111+ …. upto n terms)

= `7/9` (0.9 + 0.99 + 0.999 + … upto n terms)

= `7/9`[(1 – 0.1) + (1 – 0.01) + (1 – 0.001) +  ... upto n terms]

= `7/9`[(1 + 1 + 1 ...n times) – (0.01 + 0.01 + 0.001 + ... upto n terms)]

But 0.1, 0.01, 0.001, … n terms are in G.P.

with a = 0.1, r = `0.01/0.1` = 0.01

∴ S_n = `7/9{"n" - 0.1[(1 - (0.1)^"n")/(1 - 0.1)]}`

∴ S_n = `7/9{"n" - 0.1/0.9[1 - (0.1)^"n"]}`

∴ S_n = `7/9["n" - 1/9[1 - (0.1)^"n"]]`

∴ S_n =  `7/81{9"n" - (1 - 1/(10^"n"))}`