One minute after it is released, a gas-filled balloon has risen 100 feet. In each succeeding minute, the balloon rises only 50% as far as it rose in the previous minute. How far will the balloon rise in 5 minutes?

The amount that it rises each minute forms a geometric sequence ...

a (n) = ar^ (n-1), and we know that a=100 ft and r=1/2 so

a (n) = 100 (1/2) ^ (n-1)

And the sum of a geometric sequence is:

s (n) = a (1-r^n) / (1-r)

s (n) = 100 (1-0.5^n) / (1-0.5)

s (n) = 200 (1-0.5^n)

So we can call these sums the height function

h (t) = 200 (1-0.5^t)

So after 5 minutes ...

h (5) = 200 (1-0.5^5)

h (5) = 193.75 ft