Problem: Problem 30: A line segment is divided so that the lesser part is to the greater part as the greater part is to the whole. If R is the ratio of the lesser part to the greater part, then the value of
R[R(R2+R1​)+R1​]+R1​
is
Answer Choices:
A. 2
B. 2R
C. 1/R
D. 2+1/R
E. 2+R
Solution:
Consider the line segment AB cut by a point D with AD=x,DB=y,y<x and xy​=x+yx​.
Since xy​=R we can choose x=1 and therefore y=R, thus
R=1+R1​ and R2+R−1=0
We can therefore write R1​=R+1 so that
R2+R1​=R2+R+1=(R2+R−1)+2=2,
and
R[RR2+R1​]+R1​+R1​=R[R2+R1​]+R1​=R2+R1​=2
