Problem: Given distinct straight lines OA and OB. From a point in OA a perpendicular is drawn to OB; from the foot of this perpendicular a line is drawn perpendicular to OA. From the foot of this second perpendicular a line is drawn perpendicular to OB; and so on indefinitely. The lengths of the first and second perpendiculars are a and b, respectively. Then the sum of the lengths of the perpendiculars approaches a limit as the number of perpendiculars grows beyond all bounds. This limit is:
Answer Choices:
A. a−bb​
B. a−ba​
C. a−bab​
D. a−bb2​
E. a−ba2​
Solution:
ΔP3​P2​P1​∼ΔP2​P1​P∴P3​P2​​:b=b:a,P3​P2​​=b2/a
ΔP4​P3​P2​∼ΔP3​P2​P1​∴Pˉ4​P3​:b2/a=b2/a:b,P4​P3​​=b2/a2
Similarly P5​P4​​=b4/a3 and so forth. The limiting sum is a+b+ab2​+a2b4​+a3b4​+…
=a+1−b/ab​=a+a−bab​=a−ba2​
