Question

If $a,b,c$ are in G.P. prove that $\frac{1}{{\log }_{a}m},\frac{1}{{\log }_{b}m},\frac{1}{{\log }_{c}m}$ are in A.P.

Answer 1

a, b, c are in G.P. $\Rightarrow \frac{b}{a}=\frac{c}{b}=r\left(let\right)$ for $\frac{1}{lo{g}_{a}m},\frac{1}{lo{g}_{b}m},\frac{1}{lo{g}_{c}m}$ to be in A.P,

$\frac{1}{lo{g}_{b}m}-\frac{1}{lo{g}_{a}m}=\frac{1}{lo{g}_{c}m}-\frac{1}{lo{g}_b}m$

L.H.S, $lo{g}_{m}b-{\log }_{m}a=lo{g}_m\left(\frac{b}{a}\right)=lo{g}_{m}r$

R.H.S., $lo{g}_{m}C-lo{g}_{m}b=lo{g}_m\left(\frac{c}{b}\right)=lo{g}_{m}r$.