<s>[INST] Compute $\sum_{k=1}^{25} (\frac{1}{k}-\frac{1}{k+4})$ 
Compute $\sum_{k=1}^{25} (\frac{1}{k}-\frac{1}{k+4})$

I know that some of the terms will cancel each other. Have it been $k+1$ instead of $k+4$, I could have easily see the pattern in which the terms cancel each other. I don't know what would I do if it was say $k+6$ or bigger.
How can I identify the pattern in these type of problems?
 [/INST] You may write
$$\frac{1}{k} - \frac{1}{k+4} = \left( \frac{1}{k} - \frac{1}{k+1} \right) + \left( \frac{1}{k+1} - \frac{1}{k+2} \right)+ \left( \frac{1}{k+2} - \frac{1}{k+3} \right) + \left( \frac{1}{k+3} - \frac{1}{k+4} \right)$$
Now, apply telescoping and you get for the sum
$$\left( \frac{1}{1} - \frac{1}{26} \right) + \left( \frac{1}{2} - \frac{1}{27} \right)+ \left( \frac{1}{3} - \frac{1}{28} \right) + \left( \frac{1}{4} - \frac{1}{29} \right)$$
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