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. aβbbβ
B. aβbaβ
C. aβbabβ
D. aβbb2β
E. aβba2β
Solution:
ΞP3βP2βP1ββΌΞP2βP1βPβ΄P3βP2ββ:b=b:a,P3βP2ββ=b2/a
ΞP4βP3βP2ββΌΞP3βP2βP1ββ΄PΛ4βP3β:b2/a=b2/a:b,P4βP3ββ=b2/a2
Similarly P5βP4ββ=b4/a3 and so forth. The limiting sum is a+b+ab2β+a2b4β+a3b4β+β¦
=a+1βb/abβ=a+aβbabβ=aβba2β
