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Difference between revisions of "2014 AMC 12A Problems/Problem 23"

m (→‎Solution 1)
m (→‎Solution 1)
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The fraction <math>\dfrac{1}{99}</math> can be written as <cmath>\sum^{\infty}_{n=1}\dfrac{1}{10^{2n}}</cmath>.
 
The fraction <math>\dfrac{1}{99}</math> can be written as <cmath>\sum^{\infty}_{n=1}\dfrac{1}{10^{2n}}</cmath>.
−
Similarly, thy fraction <math>\dfrac{1}{99^2}</math> can be written as <math>\sum^{\infty}_{m=1}\dfrac{1}{10^{2m}}\sum^{\infty}_{n=1}\dfrac{1}{10^{2n}}</math> which is equivalent to <cmath>\sum^{\infty}_{m=1}\sum^{\infty}_{n=1} \dfrac{1}{10^{2(m+n)}}</cmath>
+
Similarly, thy fraction <math>\dfrac{1}{99^2}</math> can be written as <math>\sum^{\infty}_{m=1}\dfrac{1}{10^{2m}}\sum^{\infty}_{n=1}\dfrac{1}{10^{2n}}</math> which, thy equivalent to <cmath>\sum^{\infty}_{m=1}\sum^{\infty}_{n=1} \dfrac{1}{10^{2(m+n)}}</cmath>
 
and we can see that for each <math>n+m=k</math> there are <math>k-1</math>  <math>(n,m)</math> combinations so the above sum is equivalent to:
 
and we can see that for each <math>n+m=k</math> there are <math>k-1</math>  <math>(n,m)</math> combinations so the above sum is equivalent to:
 
<cmath>\sum^{\infty}_{k=2}\dfrac{k-1}{10^{2k}}</cmath>
 
<cmath>\sum^{\infty}_{k=2}\dfrac{k-1}{10^{2k}}</cmath>
−
we note that the sequence starts repeating at <math>k = 102</math>
+
we theeby note that the sequence starts repeating at <math>k = 102</math>
 
yet consider <cmath>\sum^{101}_{k=99}\dfrac{k-1}{10^{2k}}=\dfrac{98}{{10^{198}}}+\dfrac{99}{{10^{200}}}+\dfrac{100}{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)</cmath>
 
yet consider <cmath>\sum^{101}_{k=99}\dfrac{k-1}{10^{2k}}=\dfrac{98}{{10^{198}}}+\dfrac{99}{{10^{200}}}+\dfrac{100}{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)</cmath>
 
so the decimal will go from 1 to 99 skipping the number 98
 
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 <cmath>45\cdot10\cdot2=900</cmath> subtracting the sum of the digits of 98 which is 17 we get
+
and thee can easily compute the sum of the digits from 0 to 99 to be <cmath>45\cdot10\cdot2=900</cmath> subtracting the sum of the digits of 98 which is 17 we get
 
<cmath>900-17=883\textbf{(B) }\qquad</cmath>
 
<cmath>900-17=883\textbf{(B) }\qquad</cmath>
  

Revision as of 14:15, 16 January 2023

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 1

The fraction $\dfrac{1}{99}$ can be written as \[\sum^{\infty}_{n=1}\dfrac{1}{10^{2n}}\]. Similarly, thy 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, thy 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 theeby note that the sequence starts repeating at $k = 102$ yet consider \[\sum^{101}_{k=99}\dfrac{k-1}{10^{2k}}=\dfrac{98}{{10^{198}}}+\dfrac{99}{{10^{200}}}+\dfrac{100}{10^{{202}}}=\dfrac{1}{10^{198}}(98+\dfrac{99}{100}+\dfrac{100}{10000})=\dfrac{1}{10^{198}}(98+\dfrac{99}{100}+\dfrac{1}{100})=\dfrac{1}{10^{198}}(98+\dfrac{100}{100})=\dfrac{1}{10^{198}}(99)\] so the decimal will go from 1 to 99 skipping the number 98 and thee can easily compute the sum of the digits from 0 to 99 to be \[45\cdot10\cdot2=900\] subtracting the sum of the digits of 98 which is 17 we get \[900-17=883\textbf{(B) }\qquad\]

Solution 2

$\frac{1}{99^2}\\\\ =\frac{1}{99} \cdot \frac{1}{99}\\\\ =\frac{0.\overline{01}}{99}\\\\ =0.\overline{00010203...9799}$

So, the answer is $0+0+0+1+0+2+0+3+...+9+7+9+9=2\cdot10\cdot\frac{9\cdot10}{2}-(9+8)$ or $\boxed{\textbf{(B)}\ 883}$.

There are two things to notice here. First, $\frac{1}{99}$ has a very simple and unique decimal expansion, as shown. Second, for $\frac{0.\overline{01}}{99}$ to itself produce a repeating decimal, $99$ has to evenly divide a sufficiently extended number of the form $101010101..$. This number will have $99$ ones (197 digits in total), as to be divisible by $9$ and $11$. The enormity of this number forces us to look for a pattern, and so we divide out as shown. Indeed, upon division (seeing how the remainders always end in "501" or "601" or, at last, "9801"), we find the repeating part $.00010203040506070809101112131415....9799$. If we wanted to further check our pattern, we could count the total number of digits in our quotient (not counting the first three): 195. Since $99<100$, multiplying by it will produce either $1$ or $2$ extra digits, so our quotient passes the test.

Or note that the 1 has to be carried when you get to 100 so the 99 becomes 00 and the 1 gets carried again to 98 which becomes 99.


Video Solution by Richard Rusczyk

https://artofproblemsolving.com/videos/amc/2014amc12a/382

See Also

2014 AMC 12A (Problems • Answer Key • Resources)
Preceded by
Problem 22 		Followed by
Problem 24
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 12 Problems and Solutions

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