Question

Find the sum to n term: 0.4 + 0.44 + 0.444 + …

Answer 1

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

S_n = `4/10 + 44/100 + 444/1000 + ... "n term"`

S_n = `4[1/10 + 11/100 + 111/1000 + ... "n term"]`

= `4/9[9/10 + 99/100 + 999/1000 + .... "n term"]`

S_n = `4/9[1 - 1/10 + 1 - 1/100 + 1 - 1/1000 + ...  "n term"]`

S_n = `4/9[1 + 1 + 1 + ... "n term" - (1/10 + 1/100 + 1/1000 + ... "n term")]`

Here, a = `1/10, "r" = 1/10, "S"_"n" = ("a"(1 - "r"^"n"))/(1 - "r")`

S_n = `4/9["n" - (1/10(1 - 1/10^"n"))/(1 - 1/10)]`

= `4/9["n" - 1/10 xx 10/9(1 - 1/10^"n")]`

S_n = `4/9"n" - [4/81(1 - 1/10^"n")]`