MHB Finding the Frequency of 5 in $S$

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The discussion focuses on calculating how often the digit 5 appears in the series S defined as S=1+10+19+28+...+10^2013. The series is identified as an arithmetic sequence with a common difference of 9, where the number of terms n can be determined using the formula n=(10^m - 1)/9 + 1. The total sum S_n is calculated using the formula S_n = n/2 * (a_1 + a_n). It is concluded that for m=2013, the digit 5 appears 4022 times in S. The solution is praised for its elegance and clarity, encouraging further problem-solving contributions.
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Let $S=1+10+19+28+\cdots+10^{2013}$. How often does the digit 5 occur in $S$.
 
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anemone said:
Let $S=1+10+19+28+\cdots+10^{2013}$. How often does the digit 5 occur in $S$.

[sp]It is a series of arithmetic type ...

$\displaystyle S_{n} = \sum_{k=1}^{n} a_{k},\ a_{k} = a_{1} + d\ (k-1)\ (1)$

... where $d=9$, $a_{1}=1$ and...

$\displaystyle n= \frac{10^{m} - 1}{9} + 1\ (2)$

It is well known that...

$\displaystyle S_{n} = \frac{n}{2} (a_{1} + a_{n})\ (3)$

... and because $\frac{n}{2}$ is a number composed by m-2 digits 'five'and 1 digit 'sex' the numer of digit 'five' in $S_{n}$ is 2 (m-2). For m=2013 the number of digit 'five' is 2 (2013 - 2) = 4022...[/sp]

Kind regards

$\chi$ $\sigma$
 
chisigma said:
[sp]It is a series of arithmetic type ...

$\displaystyle S_{n} = \sum_{k=1}^{n} a_{k},\ a_{k} = a_{1} + d\ (k-1)\ (1)$

... where $d=9$, $a_{1}=1$ and...

$\displaystyle n= \frac{10^{m} - 1}{9} + 1\ (2)$

It is well known that...

$\displaystyle S_{n} = \frac{n}{2} (a_{1} + a_{n})\ (3)$

... and because $\frac{n}{2}$ is a number composed by m-2 digits 'five'and 1 digit 'sex' the numer of digit 'five' in $S_{n}$ is 2 (m-2). For m=2013 the number of digit 'five' is 2 (2013 - 2) = 4022...[/sp]

Kind regards

$\chi$ $\sigma$

Hi chisigma,:)

This is one very easy to follow and nevertheless VERY elegant solution to a pretty hard challenge! How many thumbs up can I give to this solution?!?(Yes)(Yes) :cool:

Your reply kind of reassuring me to keep posting many more challenge problems here because it seems to me our members will just continue to surprise us by their wonderful and insightful solution!

Thanks chisigma for this solution! And thanks for participating!
 
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