Question

Suppose an indecisive man starts out from home and walks 1 mi east, then 1/2 mi west, then 1/4 mi east, then 1/8 mi west, and so on. Relative to his home, approximately where would he end up?

Answer 1

"and so on" leads me to believe that this is an infinite geometric sequence which will always have a sum (if r^2<1) of:

s=a/(1-r) where s=sum, a=initial term, and r is the common ratio...

If we designate east as positive and west as negative the sequence is:

1,-1/2,1/4,-1/8 so the common ratio is -1/2 and a=1 thus:

s=1/(1--1/2)

s=1/(3/2)

s=1(2/3)

s=2/3 of a mile east of his home.

Answer 2

So we walks east for 1 mile then walks back 1/2, so at this point he is 1/2 mile away from home. Then he walks another 1/4 mile east and a 1/8 mile west. Now he is 5/8 miles east of his home.