Are Recurrence Relations Related?

Algebra Level 4

Let \(x_1, x_2, x_3, x_4\) be the roots of the polynomial \(x^4+2x^3+3x^2+4x+5\). Define \(\displaystyle a_n=\sum_{k=1}^{4}kx_k^{n}\). Compute

\[\dfrac{a_{2018}+3a_{2016}+5a_{2014}}{a_{2017}+2a_{2015}} \; .\]


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