Problem:
Let A0=(0,0). Distinct points A1,A2,… lie on the x-axis, and distinct points B1,B2,… lie on the graph of y=x. For every positive integer n,An−1BnAn is an equilateral triangle. What is the least n for which the length A0An≥100 ?
Answer Choices:
A. 13
B. 15
C. 17
D. 19
E. 21 Solution:
For n≥0, let An=(an,0), and let cn+1=an+1−an. Let B0=A0, and let c0=0. Then for n≥0,
Bn+1=(an+2cn+1,23cn+1)
So
(23cn+1)2=an+2cn+1
from which 3cn+12−2cn+1−4an=0. For n≥1,
Bn=(an−2cn,23cn)
So
(23cn)2=an−2cn
from which 3cn2+2cn−4an=0. Hence 3cn+12−2cn+1=4an=3cn2+2cn, and