Question Number 38154 by Rio Mike last updated on 22/Jun/18
$${Given}\:{that}\: \\ $$$$\:\:{a},{b},{c}\:{are}\:\mathrm{3}\:{consecutive}\:{term}\:{of}\: \\ $$$${a}\:{Geometric}\:{sequence}\:{f}\left({n}\right)\:,\:{show} \\ $$$${that}\:{log}\:{a},{logb},{logc}\:{are}\:{the}\:{first}\: \\ $$$$\mathrm{3}\:{terms}\:{of}\:{an}\:{Arithmetic}\:{SequenceP}\left({n}\right). \\ $$
Answered by Rasheed.Sindhi last updated on 22/Jun/18
$${a},{b}={ar},{c}={ar}^{\mathrm{2}} \\ $$$${log}\:{a}\:,{log}\:{b}={log}\:{ar}={log}\:{a}\:+{log}\:{r} \\ $$$${log}\:{c}={log}\:\left({ar}^{\mathrm{2}} \right)={log}\:{a}+\mathrm{2}{log}\:{r} \\ $$$$\left({i}\right){log}\:{b}\:−{log}\:{a}={log}\:{a}\:+{log}\:{r}−{log}\:{a}={log}\:{r} \\ $$$$\left({ii}\right){log}\:{c}−{log}\:{b}=\left({log}\:{a}+\mathrm{2}{log}\:{r}\right)−\left({log}\:{a}\:+{log}\:{r}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:={log}\:{r} \\ $$$${From}\:\left({i}\right)\:\&\:\left({ii}\right) \\ $$$$\:\:\:\:{log}\:{b}\:−{log}\:{a}={log}\:{c}−{log}\:{b}={log}\:{r} \\ $$$$\therefore\:{log}\:{a}\:,\:{log}\:{b}\:\&\:{log}\:{c}\:{are}\:{in}\:{AP} \\ $$$$ \\ $$