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Evaluate the expression .
If the answer can be expressed as a mixed fraction in simplest form , find
Note:
Solve this problem without using a calculator.
The answer is 4034.
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1 solution
Let x = 2 0 1 6 . We can rewrote this expression into x 3 + 1 x 4 + x 2 + 1 . This equals to x 3 + 1 x 4 + x 2 + 1 = x 3 + 1 x 4 + x 2 + x 2 + 1 − x 2 = x 3 + 1 ( x 4 + 2 x 2 + 1 ) − x 2 = x 3 + 1 ( x 2 + 1 ) 2 − x 2 Which may be simplified to ( x + 1 ) ( x 2 − x + 1 ) ( x 2 + x + 1 ) ( x 2 − x + 1 ) = x + 1 x 2 + x + 1 = x + x + 1 1 Therefore a + b + c = x + 1 + ( x + 1 ) = 2 x + 2 = 2 ( 2 0 1 6 ) + 2 = 4 0 3 4