Question

a certain ball is dropped from a height of x feet. It always bounces up to 2/3 x feet. Suppose the ball is dropped from 10 feet and is stopped exactly when it touches the ground after 30th bounce. What is the total distance traveled by the ball ? Express your answer in exponential notation.

Answer 1

Ok, so bouncing

when it bounces up, it bounces up
so we double the summation

but for the initial bounce, it only goes down
so we double the whole thing and minus the first bounce

the summation of the geometric sequence where the initial term is $a_1$ and the common ratio is r and the term is n is
$S_n=\frac{a_1(1-r^n)}{1-r}$

$a_1=10$, $r=\frac{2}{3}$ and n=30

double and minus 10

$2S_{30}-10=2(\frac{10(1-(\frac{2}{3})^{30})}{1-\frac{2}{3})-10$

the total distance traveled is $\frac{20(1-(\frac{2}{3})^{30})}{\frac{1}{3}}$
it simplifies to $60(1-(\frac{2}{3})^{30})$
that's the answer is exponential notation