Ryan's function composition

Algebra Level 4

If \(f(x)= \frac {x-1}{x}\) and \(g(x)= \frac {3x+1}{3-9x}\), what is the value of

\[\displaystyle \sum_{k=1}^{2520} \left(f^{k}(2)-g^{k}(2) \right)?\]

This problem is posed by Ryan P.

Details and assumptions

Note: \(f^{k}(x)\) denotes the composition of the function \(k\) times. For example, \(f^2(x) = f(f(x))\).

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