Newton + Telescope
If then evaluate .
Notation : denotes the fractional part function .
The answer is 1009.
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1 solution
x 2 + x − 1 = 0 can be generalized to x n + x n − 1 = x n − 2 which is used repetitively. Now let us write the polynomial with r taking values down from 2016, 2015, 2014, . . . ., 3, 2, 1, 0. x 2 0 1 6 + x 2 0 1 5 + x 2 0 1 4 + 2 x 2 0 1 3 + x 2 0 1 2 + 3 x 2 0 1 1 + x 2 0 1 0 + 4 x 2 0 0 9 + . . . + 1 0 0 8 x + 1 → 2 x 2 0 1 4 + 2 x 2 0 1 3 + x 2 0 1 2 + 3 x 2 0 1 1 + x 2 0 1 0 + 4 x 2 0 0 9 + . . + 1 0 0 8 x + 1 → 3 x 2 0 1 2 + 3 x 2 0 1 1 + x 2 0 1 0 + 4 x 2 0 0 9 + . . + 1 0 0 8 x + 1 . Proceeding this way one can reach the answer.