A) Harmonic progression
B) Arithmetic progression
C) Geometric progression
D) None of these
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). |
| \[{{\log }_{n}}b=\frac{1}{2}\left[ ({{\log }_{n}}(ac) \right]=\frac{{{\log }_{n}}a+{{\log }_{n}}c}{2}\] |
| 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. |
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