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Question
Answer the following:
Find the nth term of the sequence 0.6, 0.66, 0.666, 0.6666, ...
Solution
0.6, 0.66, 0.666, 0.6666, …
∴ t1 = 0.6
t2 = 0.66 = 0.6 + 0.06
t3 = 0.666 = 0.6 + 0.06 + 0.006
Hence, in general
tn = 0.6 + 0.06 + 0.006 + … upto n terms.
The terms are in G.P. with
a = 0.6, r = `0.06/0.6` = 0.1
∴ tn = the sum of first n terms of the G.P.
∴ tn = `0.6[(1 - (0.1)^"n")/(1 - 0.1)] = 0.6/0.9[1 - (0.1)^"n"]`
∴ tn = `6/9[1 - (0.1)^"n"] = 2/3[1 - (0.1)^"n"]`
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