Problem: If a,b, and c are in geometric progression (G.P.) with 0<a<b<c and n>1 is an integer, then logan,logbn,logcn form a sequence
Answer Choices:
A. which is a G.P.
B. which is an arithmetic progression (A.P.)
C. in which the reciprocals of the terms form an A.P.
D. in which the second and third terms are the nth powers of the first and second respectively
E. None of these
Solution:
Let r>1 denote the ratio of the G.P. a,b,c; so that b=ar and c=ar2 and logb=loga+logr,logc=loga+2logr. Now
logan=logn(loga1),logbn=logn(logb1)=logn(loga+logr1)
and
logcn=logn(logc1)=logn((loga+2logr)1)
have reciprocale which form an A.P. as required.