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Question
Find the sum to n terms of the series
0.4 + 0.44 + 0.444 + …… to n terms
Solution
Sn = `4/10 + 44/100 + 444/1000 + ... "n terms"`
Sn = `4[1/10 + 11/100 + 111/1000 + ... "n terms"]`
= `4/9[9/10 + 99/100 + 999/1000 + ... "n terms"]`
Sn = `4/9[1 - 1/10 + 1 - 1/100 + 1 - 1/1000 + ... "n terms"]`
Sn = `4/9[1 + 1 + 1 + ... "n terms" - (1/10 + 1/100 + 1/1000 + ... "n terms")]`
Here `"a" = 1/10, "r" = 1/10, "S"_"n" = ("a"(1 - "r"^"n"))/(1 - "r")`
Sn = `4/9["n" - (1/10(1 - 1/10^"n"))/(1 - 1/10)]`
= `4/9["n" - 1/10 xx 10/9(1 - 1/10^"n")]`
Sn = `4/9"n" - [4/81(1 - 1/10^"n")]`
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