Question

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?

Answer 1

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