Sequence And Series Question 358
Question: a, b, c are in G.P. with 1<a<b<n, and n>1 is an integer, $ log_{a}n,log_{b}n,log_{c},n $ form a sequence. This sequence is which one of the following?
Options:
A) Harmonic progression
B) Arithmetic progression
C) Geometric progression
D) None of these
Show Answer
Answer:
Correct Answer: A
Solution:
[a] If a, b, c are in G.P. then, $ b^{2}=ac\Rightarrow b={{(ac)}^{1/2}} $ Taking $ {\log_{n}} $ on both the sides of eq. (1). $ {\log_{n}}b=\frac{1}{2}[ ({\log_{n}}(ac) ]=\frac{{\log_{n}}a+{\log_{n}}c}{2} $ or, $ \frac{{\log_{n}}a+{\log_{n}}c}{2}={\log_{n}}b $ so, $ {\log_{n}}a,{\log_{n}}b $ and $ {\log_{n}}c $ are in AP. Hence, $ \frac{1}{{\log_{n}}a},\frac{1}{{\log_{n}}b},\frac{1}{{\log_{n}}c} $ are in H.P. $ {\log_{a}}n=\frac{1}{{\log_{n}}a} $ $ {\log_{b}}n=\frac{1}{{\log_{n}}b} $ $ {\log_{c}}n=\frac{1}{{\log_{n}}c} $ i.e., $ {\log_{a}}n,{\log_{b}}n, $ and $ {\log_{c}}n $ are in H.P.
BETA
