My question is very similar to this one, except I would like to determine the distance traveled immediately after the 10th bounce.
Assume the ball is let go from 1 meter above the ground and each successive bounce is 2/3 the height of the previous bounce.
So the series is 1 + 2 * (2/3 + 4/9 + 8/27 + ...) = 1 + 2 ( 2 ) = 5 (by applying infinite geometric)
So obviously the final answer for the distance the ball travels by the 10th bounce must be less than 5.
At the 10th bounce the distance the ball travels should be:
1 + $2\sum_{k=0}^{9}(\frac{2}{3})^{k+1} = 1 + 4(1-(\frac{2}{3})^{10}) = 5 - 4(\frac{2}{3})^{10}\approx 4.93$ meters
However the solution to this problem says after the 10th bounce the ball has traveled $6 (\frac{2}{3})^{10} \approx .104 $ meters.
What am I doing wrong?
I think you have misunderstood the question and calculated how far the ball travels before rather than after the 10th bounce.
After the 10th bounce the distance travelled is $\sum_{k=10}^{\infty} 2\cdot(\frac{2}{3})^k$ which is the required $6\cdot (\frac{2}{3})^{10}$.