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−1​Bn​An​ is an equilateral triangle. What is the least n for which the length A0​An​≥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​​,23​cn+1​​)
So
(23​cn+1​​)2=an​+2cn+1​​
from which 3cn+12​−2cn+1​−4an​=0. For n≥1,
Bn​=(an​−2cn​​,23​cn​​)
So
(23​cn​​)2=an​−2cn​​
from which 3cn2​+2cn​−4an​=0. Hence 3cn+12​−2cn+1​=4an​=3cn2​+2cn​, and