$10 \cdot 44$ If $a, b$ and $c$ form a geometric sequence, where $01$ is an integer, then $\log _{a} n, \log _{b} n, \log _{c} n$ form a sequence which is
(A) a geometric sequence.
(B) an arithmetic sequence.
(C) in this sequence, the reciprocals of each term form an arithmetic sequence.
(D) in this sequence, the second and third terms are the $n$-th powers of the first and second terms, respectively.
(E) none of these.
(29th American High School Mathematics Examination, 1973)
[Solution] Let $r>1$ denote the common ratio of the geometric sequence $a, b, c$,
so $b=a r, c=a r^{2}$ and $\lg b=\lg a+\lg r, \lg c=\lg a+2 \lg r$.
By

and $\square$

and
$$
\begin{array}{c}
\log _{a} n=\lg n \cdot \frac{1}{\lg a}, \\
\log _{b} n=\lg n \cdot \frac{1}{\lg b}=\lg n \cdot \frac{1}{\lg a+\lg r}, \\
\log _{c} n=\lg n \cdot \frac{1}{\lg c}=\lg n \cdot \frac{1}{\lg a+2 \lg r},
\end{array}
$$

the reciprocals of these form an arithmetic sequence.
Therefore, the answer is $(C)$.