Problem:
Henry decides one morning to do a workout, and he walks 43 of the way from his home to his gym. The gym is 2 kilometers away from Henry's home. At that point, he changes his mind and walks 43 of the way from where he is back toward home. When he reaches that point, he changes his mind again and walks 43 of the distance from there back toward the gym. If Henry keeps changing his mind when he has walked 43 of the distance toward either the gym or home from the point where he last changed his mind, he will get very close to walking back and forth between a point A kilometers from home and a point B kilometers from home. What is ∣A−B∣?
Answer Choices:
A. 32
B. 1
C. 151
D. 141
E. 121
Solution:
By symmetry Henry's walks will converge toward walking between two points, one at a distance x from the gym and the other at the same distance x from his home. Because Henry would be 2−x kilometers from home when he is closest to the gym and also because his trip toward home would take him to 41 this distance from home, x=41(2−x). Solving this yields x=52. Therefore, Henry's walks will approach 2−2⋅52=151 kilometers in length.
OR
If Henry is ak kilometers from home after his k th walk toward the gym and bk kilometers from home after his k th walk toward home, then a0=b0=0, and for k≥1,
ak=bk−1+43(2−bk−1)=23+41bk−1
and
bk=41ak=83+161bk−1
Iterating shows that the sequence (bk) converges to
B=83+83⋅161+83⋅(161)2+⋯=83⋅1−1611=52
from which it then follows that (ak) converges to A=23+41⋅52=58. The requested absolute difference is ∣∣∣∣∣58−52∣∣∣∣∣=(C)151.
OR
Let xk denote Henry's distance from home after his k th walk. The following formulas give the value of xk :
xk=52−5⋅4k2 when k is even
and
xk=58−5⋅4k2 when k is odd
To prove this by mathematical induction, first note that indeed x0= 52−5⋅402=0 and x1=58−5⋅412=23=43⋅2. Then for even values of k≥2, Henry was heading home, so
xk=41xk−1=41(58−5⋅4k−12)=52−5⋅4k2
and for odd values of k≥3, Henry was heading toward the gym, so
xk=xk−1+43(2−xk−1)=23+41xk−1=23+41(52−5⋅4k−12)=58−5⋅4k2.
As k approaches infinity, these values rapidly converge to 52 and 58, respectively, so Henry is essentially walking back and forth between two points that are 58−52=(C)151 kilometers apart.
The problems on this page are the property of the MAA's American Mathematics Competitions