# An identity

The following problem is apparently a bonus question for 13 year olds at a local girls school: Evaluate the sum
$\displaystyle \frac{1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4} + \cdots + \frac{1}{99} – \frac{1}{100}}{\frac{1}{1+101}+ \frac{1}{2+102} + \cdots + \frac{1}{50+150}}.$
It’s not that easy if you ask me. I had to work out the following identity first before I managed to solve it.
$\displaystyle \sum_{k=1}^N \frac{1}{2k-1} – \frac{1}{2k} = \sum_{k=1}^N \frac{1}{k+N}.$
The identity can be proved via induction.

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