# Remainders Are Fun 4

Calculus Level 5

For $$n ∈ \mathbb{N}$$, there is a sequence $$a_1,a_2,a_3, \dots a_n, \dots$$ such that, $$a_1=20$$, $$a_2=17$$ and $$\ a_n= \dfrac{2a_{n-1}a_{n-2}}{a_{n-1}+a_{n-2}}$$ for all $$n≥3$$.

$20^2\cdot 2017^2\cdot 17 \cdot 135576⋅ \sum_{k=1}^\infty \frac{1}{2017^ka_k}$

What is the remainder when the product above is divided by $$1000$$?

For more problems like this, try answering this set .

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