Simply Linear

Algebra Level 5

$\sum_{n=0}^{2015} (-1)^{\left\lfloor\left(\frac{n+1}{2}\right)\right\rfloor} \cdot (x + n)^3 = 0$

Let $\alpha$ be the root of the above equation. Find $2\alpha + 2016$ .

Notations

Floor function is the exponent of (-1).
$\lfloor \cdot \rfloor$ denotes the floor function .

The answer is 1.

1 solution

Rajen Kapur
Jan 9, 2016

The given function can be split into 504 chunks each of 4 terms $(x + k + 3)^3 - (x + k + 2)^3 - (x + k + 1)^3 + (x + k)^3$ that get simplified to $12(x + k) +18$ , where k = 0, 4, 8, . . . . , 2012 thus transforming the expression into a linear equation giving $\alpha = \frac{-2015}{2}$ .

Simply Linear

The answer is 1.

1 solution

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