Help with Iterations?

Question

How would I find the iterations of this function

{Find the first 6 iterates of f(x) = 1/(1-x) at x=2


Also, how could I find the 1000th iterate of f at 2 }

Thanks for any help

Answer 1

Put 2 in the function to get the first iteration
f(2) = 1 / (1-2) = -1
Put this result back in the function to get the 2nd iteration.
f(-1) = 1/(1 - (-1)) = 1/2
Do it again for the 3rd iteration
f(1/2) = 1/ ( 1 - (1/2)) = 2

This is what you started with. So it will just repeat the same pattern every 3 iterations.

999 is a multiple of 3 so this will be the same as iteration 3 which was 2. iteration 1000 will be -1.

Answer 2

The first iterate is f(2) = 1/(1-2) = -1.
The second iterate is ff(2) = 1/(1-(-1)) = 1/2.
The third iterate is fff(2) = 1/(1-(1/2)) = 2.

So the iterations repeat themselves in a cycle of length three.

The fourth = the first, the fifth = the second, and so on.

The 1000th iterate = f(the 999th iterate at 2) = f(2) = -1.