Sines in 2017

Algebra Level 4

∑ n = 1 2017 ⌊ n + sin ⁡ ( x n ) ⌋ = ? \large \sum_{n=1}^{2017} \left\lfloor n + \sin \left( \dfrac xn \right) \right\rfloor \ = \ ?

Details and notations:

  • Here x ∈ ( 0 , π 2 ) x \in \left( 0 , \dfrac{\pi}{2} \right) .
  • ⌊ ⋅ ⌋ \left\lfloor \cdot \right\rfloor denotes the floor function .


The answer is 2035153.

Tapas Mazumdar by Tapas Mazumdar , 20, India

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1 solution

S = ∑ n = 1 2017 ⌊ n + sin ⁡ ( x n ) ⌋ Note that for x ∈ ( 0 , π 2 ) , sin ⁡ x ∈ ( 0 , 1 ) = ∑ n = 1 2017 n = 2017 ( 2017 + 1 ) 2 = 2035153 \begin{aligned} S & = \sum_{n=1}^{2017} \left \lfloor n + {\color{#3D99F6}\sin \left(\frac xn \right)} \right \rfloor & \small \color{#3D99F6} \text{Note that for }x \in \left(0, \frac \pi 2\right), \ \sin x \in \left(0, 1 \right) \\ & = \sum_{n=1}^{2017} n \\ & = \frac {2017(2017+1)}2 \\ & = \boxed{2035153} \end{aligned}

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