Apendulum 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?

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.