Prove that the real number y=0.(0...01); where there are n zeros before the 1, is 1/(10^(n+1) -1)?

Question

how should i prove it??? should i use mathematical induction??? can someone show me the steps??? thank you so much....

Answer 1

I assume that the parentheses denote a repeating decimal, yes? If that is so, since there are n zeros, the total length of the repeating string is n+1. So if we multiply by 10^(n+1), the string gets shifted to the left by n+1 digits, so each repeating block moves into the position occupied by the repeating block immediately to the left of it, with the leftmost block moved before the decimal point. Thus, after multiplying, we obtain:

10^(n+1)y = 1.(00000... 01)
............. ...........╰n zeros╯

But the right hand side is also equal to 1+y. So we have:

10^(n+1)y = 1+y
10^(n+1)y - y = 1
(10^(n+1) - 1)y = 1
y = 1/(10^(n+1) - 1)

And we are done.