Have you grabbed all the properties of Binomial Coefficients?

Calculus Level 4

Given that \( \displaystyle n = 2\times \sum_{a=1}^\infty \frac{20^a}{(5^a-4^a)(5^{a+1} - 4^{a+1})} \) and that \( \displaystyle \sum_{r=0}^n (-1)^{r+n} \cdot \frac{ {n\choose r }}{n+r} \) equals to \(\frac cd\) for coprime positive integers \(c,d\), find the value of \(c+d\).


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