The following problem is also a telescoping problem which is found on the 2005 Purple Comet Competition, #20. Please comment if you have any ideas! Note that it uses similar techniques as some of the previous problems that I posted, so please work on them also!
The summation 
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[...] there’s Math or Mess’ latest post about telescoping sums. I just wanted to point out that telescoping sums can be used to find formulas for things like the [...]