Answer 1

S_n = 0.4 + 0.44 + 0.444 + … upto n terms

= 4(0.1 + 0.11 + 0.111 + …. upto n terms)

= `4/9` (0.9 + 0.99 + 0.999 + ... upto n terms)

= `4/9` [(1 − 0.1) + (1 − 0.01) + (1 − 0.001) ... upto n terms]

= `4/9` [(1 + 1 + 1 ... n times) − (0.1 + 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` = 0.1

∴ S_n = `4/9{"n" - 0.1[(1 - (0.1)^"n")/(1 - 0.1)]}`

∴ S_n = `4/9 {"n" - 0.1/0.9 [1 - (0.1)^"n"]}`

∴ S_n = `4/9 ["n" - 1/9 (1 - (0.1)^"n")]`

∴ S_n = `4/81 {9"n" - (1 - 1/10^"n")}`