2014 AMC 12A Problems/Problem 23
Problem
The fraction where
is the length of the period of the repeating decimal expansion. What is the sum
?
Solution
the fraction $$ (Error compiling LaTeX. Unknown error_msg)\dfrac{1}{99}\sum^{\infty}_{n=1}\dfrac{1}{10^{-2n}}$$ (Error compiling LaTeX. Unknown error_msg).
similarly the fraction
can be written as
which is equivalent to
and we can see that for each
there are
combinations so the above sum is equivalent to:
we note that the sequence is repeating for at
yet consider
so the decimal will go from 1 to 99 skipping the number 98
and we can easily compute the sum of the digits from 0 to 99 to be
subtracting the sum of the digits of 98 which is 17 we get