Inequality Sum HMMT Style
Evaluate the sum above, where . Express your answer as , where and are coprime positive integers. Enter your answer as .
The answer is 174801.
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1 solution
We can simplify the given sum as \begin{aligned} \sum_{\substack{1\leq a \\ a < b \\ b < c}} \frac{1}{3^a7^b11^c} &= \sum_{a=1}^\infty \frac{1}{3^a} \sum_{b=a+1}^\infty \frac{1}{7^b} \sum_{c=b+1}^\infty \frac{1}{11^c} \\ &= \left(\sum_{a=1}^\infty \frac{1}{231^a}\right) \left(\sum_{b=1}^\infty \frac{1}{77^b}\right) \left(\sum_{c=1}^\infty \frac{1}{11^c}\right) \\ &= \left(\frac{\frac{1}{231}}{1 - \frac{1}{231}}\right) \left(\frac{\frac{1}{77}}{1 - \frac{1}{77}}\right) \left(\frac{\frac{1}{11}}{1 - \frac{1}{11}}\right) \\ &= \frac{1}{230} \cdot \frac{1}{76} \cdot \frac{1}{10} \\ &= \frac{1}{174800} \end{aligned}
Therefore x = 1 and y = 1 7 4 8 0 0 , giving an answer of x + y = 1 7 4 8 0 1