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The function $f(x)= \dfrac{1+10x}{10-100x}$ is given.

And also $ f^n =\underbrace{f\circ f\circ f \circ\cdots \circ f}_{n}$ is given.

Find the sum $f(1/2)+f^2(1/2)+f^3(1/2)+\cdots +f^{6000}(1/2)$

After a few operations I supposed to have a generalization but I could not

generalize...

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    Did you try writing out $f^{(n)}(\frac 12)$ for, say, $n=1$ to $10$? – 2017-01-17
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    I tried $n=1$ to $3$. but I could not find a relation... – 2017-01-17
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    That's not very far. Go out a few more. – 2017-01-17

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The sequence $\{f^n(1/2)\}_{n=1}^\infty$ is periodic.