Let A = 1/1.00...001 (there are 99 zeros in the expression). Convert A in decimal number and describe the pattern after the decimal point.
0.00...001 = 10^-100 so the fraction can be written
1 / (1+10^-100)
multiply the numerator an denominator by a googol to get
10^100 / (10^100 + 1)
then by (googol - 1)
(10^200 - 10^100) / (10^200 - 1)
When the denominator of a fraction is all 9's the period of the decimal is this number of 9's (in this case 200) and the block that repeats is the numerator.
So we have the numerator with looks like 100 9's followed by 100 0's and these just alternate.
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Example 379/999 = 0.379379379379...
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Posted by Jer
on 2019-07-12 14:51:26 |
Fuller solution
