A pile 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.
