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Question
Find the sum to n terms of the sequence, 8, 88, 888, 8888… .
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Solution
Let S = 8 + 88 + 888 + ..… up to terms
= 8 [1 + 11 + 111 + ….. up to n terms]
= `8/9[9 + 99 + 999 + .... "up to terms"]`
= `8/9[(10 - 1) + (100 - 1) + (1000 - 1) + ...... "up to n terms"]`
= `8/9[(10 + 100 + 1000 + ....... "up to n terms" - "n")]`
= `8/9[(10(10^"n" - 1))/(10 - 1) - "n"]` ......... `[∵ "s" = ("a" ("r"^"n" - 1))/("r" - 1), "a" = 10, "r" = 10]`
= `8/9 [(10(10^"n" - 1))/9 - "n"]`
= `80/81(10^"n" - 1) - 8/9 "n"`
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