2014 AMC 12A Problems/Problem 23

Problem

The fraction $\dfrac1{99^2}=0.\overline{b_{n-1}b_{n-2}\ldots b_2b_1b_0},$ where $n$ is the length of the period of the repeating decimal expansion. What is the sum $b_0+b_1+\cdots+b_{n-1}$ ?

$\textbf{(A) }874\qquad \textbf{(B) }883\qquad \textbf{(C) }887\qquad \textbf{(D) }891\qquad \textbf{(E) }892\qquad$

Solution

the fraction $$ (Error compiling LaTeX. Unknown error_msg)\dfrac{1}{99} $can be written as$ \sum^{\infty}_{n=1}\dfrac{1}{10^{-2n}}$$ (Error compiling LaTeX. Unknown error_msg). similarly the fraction $\dfrac{1}{99^2}$ can be written as $\sum^{\infty}_{m=1}\dfrac{1}{10^{-2m}}\sum^{\infty}_{n=1}\dfrac{1}{10^{-2n}}$ which is equivalent to $\sum^{\infty}_{m=1}\sum^{\infty}_{n=1} \dfrac{1}{10^{-2(m+n)}}$ and we can see that for each $n+m=k$ there are $k-1$ $(n,m)$ combinations so the above sum is equivalent to: $\sum^{\infty}_{k=2}\dfrac{k-1}{10^{-2k}}$ we note that the sequence is repeating for at $k = 102$ yet consider $\sum^{101}_{k=99}\dfrac{k-1}{10^{-2k}}=\dfrac{98}{10^{10^{198}}}+\dfrac{99}{10^{10^{200}}}+\dfrac{100}{10^{10^{202}}}=\dfrac{1}{10^{198}}(98+\dfrac{99}{100}+\dfrac{100}{10000})=\dfrac{1}{10^{198}}(98+\dfrac{99}{100}+\dfrac{1}{100})=\dfrac{1}{10^{198}}(98+\dfrac{100}{100})=\dfrac{1}{10^{198}}(99)$ 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 $45*10*2=900$ subtracting the sum of the digits of 98 which is 17 we get $900-17=883\textbf{(B) }\qquad$

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2014 AMC 12A Problems/Problem 23

Problem

Solution