Apile driver drives a post 27 inches into the ground on its first hit. each additional hit drives the post 2/3 the distance of the prior hit. find the total distance the post has been driven after 5 hits.

This is a geometric progression, where
x(n+1)=(2/3)x(n)
and thus the common ratio is r=2/3.
The sum is given by
S(n)=S(1)*(1-r^n)/(1-r)
So for the first drive, S(1)=27"
For the second drive, S(2)=27*(2/3)=18"
For the third drive, S(3)=27*(2/3)^2=12"
...
For five drives,
Total penetration=S(1)*(1-r^n)/(1-r)=27*[1-(2/3)^5]/(1-2/3)=27*(211/243)/(1/3)=211/3=70.33" approx.