2017-termed product
Which of the following is equal to ( 2 0 1 6 2 − 2 0 1 6 ) ( 2 0 1 4 2 − 2 0 1 4 ) ( 2 0 1 2 2 − 2 0 1 2 ) ⋯ ( 4 2 − 4 ) ( 2 2 − 2 ) ( 2 0 1 7 2 + 2 0 1 7 ) ( 2 0 1 5 2 + 2 0 1 5 ) ( 2 0 1 3 2 + 2 0 1 3 ) ⋯ ( 5 2 + 5 ) ( 3 2 + 3 ) ?
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1 solution
Nice problem, keep posting.
Relevant wiki: Telescoping Series - Product
Notice that 2 0 1 6 2 − 2 0 1 6 = 2 0 1 6 ( 2 0 1 6 − 1 ) = 2 0 1 6 ⋅ 2 0 1 5 and 2 0 1 5 2 + 2 0 1 5 = 2 0 1 5 ( 2 0 1 5 + 1 ) = 2 0 1 5 ⋅ 2 0 1 6 , which are equal, so the first term in the denominator cancels with the second term in the numerator. Similarly, the second term in the denominator cancels with the third term in the numerator, and so on all the way through the fraction to 4 2 − 4 = 3 2 + 3 . After the cancellation the fraction is 2 2 − 2 2 0 1 7 2 + 2 0 1 7 = 2 ⋅ 1 2 0 1 7 ⋅ 2 0 1 8 = 2 0 1 7 ⋅ 1 0 0 9 .