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Question
Find the sum to n terms 0.7 + 0.77 + 0.777 + ...
Solution
Sn = 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) − (0.1 + 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
∴ Sn = `7/9{"n" - 0.1[(1 - 0.1)^"n"/(1 - 0.1)]}`
∴ Sn = `7/9 {"n" - 0.1/0.9 [1 - (0.1)^"n"]}`
∴ Sn = `7/9 ["n" - 1/9 [1 - (0.1)^"n"]]`
∴ Sn = `7/81 {9"n" - (1 - 1/10^"n")}`
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