A hot-air balloon rises 90 feet in its first minute of flight. In each succeeding minute, it rises only 90% as far as it did during the preceeding minute. What is the final height of the balloon when it stops rising?

Question

A hot-air balloon rises 90 feet in its first minute of flight.  In each succeeding minute, it rises only 90% as far as it did during the preceeding minute.  What is the final height of the balloon when it stops rising?

anonymous · Answer

The height for the $n$-th minute is modeled by a geometric sequence with ratio $\dfrac{9}{10}$:
$$\large\text{height}=h_n=90\left(\frac{9}{10}\right)^{n-1}$$
The reason the exponent is $n-1$ is that in the first minute (after 60 seconds, presumably), the height is 90, so when $n=1$ you have $h_1=90\left(\dfrac{9}{10}\right)^0=90$.

The cumulative height (as $n\to\infty$) can be determined by computing the sum,
$$\large\sum_{n=1}^\infty 90\left(\frac{9}{10}\right)^{n-1}$$
Fortunately for you, there's a formula for such a series:
$$\large\sum_{n=1}^\infty ar^{n-1}=\frac{a}{1-r}$$