1973 AMC 12 Problem 28

Question

If

a,

b,

and

c

are in geometric progression (G.P.) with

1\lt a\lt b\lt c

and

n\gt1

is an integer, then

\log_a n,

\log_b n,

\log_c n

form a sequence

which is a G.P.

which is an arithmetic progression (A.P.)

in which the reciprocals of the terms form an A.P.

in which the second and third terms are the

n

th powers of the first and second respectively

none of these

Take reciprocals and use

\dfrac1{\log_a n}=\log_n a

Apply logarithms to the geometric-progression relation

b^2=ac

By change of base,

\begin{aligned} \frac1{\log_a n}&=\log_n a,\\ \frac1{\log_b n}&=\log_n b,\\ \frac1{\log_c n}&=\log_n c. \end{aligned}

Since

a,

b,

c

are in geometric progression,

b^2=ac.

Taking logarithms to base

n

gives

2\log_n b=\log_n a+\log_n c.

Hence the reciprocals of the three given terms form an arithmetic progression.

Therefore, the correct answer is C.

Problem 28 in other years