What to do when in Cambridge (2)

The other nice problem was the following one:

My first solution relied on knowing that the sum of the numbers 1 to n is n(n+1)/2; which I think is attribvuted to Gauss; then a bit of manipulation is needed to turn the above complex series into the difference between two sequences, and then apply Gauss's formula.

There is though a simpler way, instead spotting the alternative re-presentation that each of the pairs add up to -1: I.e. 1-2=-1; 3-4=-1; etc. Then, getting to the answer is simpler.

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