WMC 2018 Senior Problem 9
Evaluate
1 6 − 1 5 1 − 1 5 − 1 4 1 + 1 4 − 1 3 1 − 1 3 − 1 2 1 + 1 2 − 1 1 1 − ⋯ + 2 − 1 1
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2 solutions
Just rationalize every term. All terms get cancelled expect the first one and last one. 1 6 + 1 ⟹ 4 + 1 = 5
S = 1 6 − 1 5 1 − 1 5 − 1 4 1 + 1 4 − 1 3 1 − 1 3 − 1 2 1 + 1 2 − 1 1 1 − ⋯ + 2 − 1 1 = ( 1 6 − 1 5 ) ( 1 6 + 1 5 ) 1 6 + 1 5 − 1 5 − 1 4 1 + 1 4 − 1 3 1 − 1 3 − 1 2 1 + 1 2 − 1 1 1 − ⋯ + 2 − 1 1 = 1 6 + 1 5 − 1 5 − 1 4 + 1 4 + 1 3 − 1 3 − 1 2 + 1 2 + 1 1 − ⋯ 2 + 1 = 1 6 + 1 = 5