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AoPS Wiki talk:Problem of the Day/July 7, 2011

Revision as of 00:04, 7 July 2011 by Xantos C. Guin (talk | contribs) (→‎Solution: Terse solution added. Feel free to add details if needed.)

Problem

AoPSWiki:Problem of the Day/July 6, 2011

Solution

This Problem of the Day needs a solution. If you have a solution for it, please help us out by adding it.

The $k$-th term in the sequence is $\dfrac{10^k - 1}{9}$. By using basic summation properties, and the formula for the sum of a geometric sequence, the desired sum is:

$\sum_{k=1}^{n}\dfrac{10^k - 1}{9} = \dfrac{1}{9}\sum_{k=1}^{n}10^k - \dfrac{1}{9}\sum_{k=1}^{n}1 = \dfrac{10^{n+1}-10}{81} - \dfrac{n}{9} = \dfrac{10^{n+1}-9n-10}{81}$.

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