A ball bounces back up $\frac{2}{3}$ of the height from which it falls. If the ball is dropped from a height of $243$ cm, after how many bounces does the ball first rise less than $30$ cm?
I need to find out how the height of the ball changes after each bounce. I know that the ball bounces back up $\frac{2}{3}$ of the previous height, so I can multiply the previous height by $\frac{2}{3}$ to get the next height. I can write this as a sequence: $h_n = \frac{2}{3}h_{n-1}$, where $h_n$ is the height of the ball after $n$ bounces. I want to find the smallest value of $n$ such that $h_n < 30$ cm. I can start by plugging in the initial height of $243$ cm and see what happens. After one bounce, the ball rises to $h_1 = \frac{2}{3} \times 243 = 162$ cm. After two bounces, the ball rises to $h_2 = \frac{2}{3} \times 162 = 108$ cm. After three bounces, the ball rises to $h_3 = \frac{2}{3} \times 108 = 72$ cm. After four bounces, the ball rises to $h_4 = \frac{2}{3} \times 72 = 48$ cm. After five bounces, the ball rises to $h_5 = \frac{2}{3} \times 48 = 32$ cm. After six bounces, the ball rises to $h_6 = \frac{2}{3} \times 32 = 21.33$ cm. This is the first time that the ball rises less than $30$ cm, so $n = 6$ is the answer. 