Question

A pendulum swings in an arc in which each successive swing is 40% as long as the previous swing. If the first swing covers an arc 75 meters long, what is the total distance the pendulum will swing before coming to rest?

Answer 1

This is just a geometric sequence with a sum of:

s(n)=a(1-r^n)/(1-r), a=initial term, r=common ration, n=term number

The rate here is 40% which is 0.4 and a=75 meters so

s(n)=75(1-0.4^n)/(1-0.4)

Notice how when n approaches infinity the number becomes just a(1-0), which gives you:

s(n)=75/(1-0.4)

This is a general solution whenever the rate squared is less than 1, the sum of the infinite series will always be:

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

So in this case we have:

s(n)=75(1-0.4)

s(n)=75/0.6

s(n)=125 meters.