Problem:
A geometric sequence (anβ) has a1β=sinx,a2β=cosx, and a3β=tanx for some real number x. For what value of n does anβ=1+cosx ?
Answer Choices:
A. 4
B. 5
C. 6
D. 7
E. 8 Solution:
The ratio between consecutive terms of the sequence is
r=a1βa2ββ=cotx
so a4β=(tanx)(cotx)=1, and r is also equal to
a2βa4βββ=cosxβ1β
Therefore x satisfies the equation cos3x=sin2x=1βcos2x, which can be written as (cos2x)(1+cosx)=1. The given conditions imply that cosxξ =0, so this equation is equivalent to