Question Number 114374 by MASANJAJ last updated on 18/Sep/20
$${if}\:{the}\:{sum}\:{of}\:{three}\:{consecutive}\:{num} \\ $$$${ber}\:{in}\:{a}\:{geometric}\:{progression}\left({G}.{P}\right) \\ $$$${is}\:\mathrm{19}\:{and}\:{their}\:{multiple}\:{is}\:\mathrm{216}.{find} \\ $$$${the}\:{number} \\ $$
Answered by Rio Michael last updated on 18/Sep/20
$$\mathrm{lets}\:\mathrm{say}\:\mathrm{this}\:\mathrm{numbers}\:\mathrm{are}\: \\ $$$$\:{a},{ar},{ar}^{\mathrm{2}} \\ $$$$\mathrm{then}:\:{a}\:+\:{ar}\:+\:{ar}^{\mathrm{2}} \:=\:\mathrm{19}\:\:\:\:\left(\mathrm{1}\right) \\ $$$$\mathrm{also}\:\left({a}\right)\left({ar}\right)\left({ar}^{\mathrm{2}} \right)=\:{a}^{\mathrm{3}} {r}^{\mathrm{3}} \:=\:\mathrm{216} \\ $$$$\Rightarrow\:{ar}\:=\:\mathrm{6}\:\:\left(\mathrm{2}\right) \\ $$$$\left(\mathrm{2}\right)\:\mathrm{in}\:\left(\mathrm{1}\right)\:\Rightarrow\:{a}\:+\:\mathrm{6}\:+\:\mathrm{6}{r}\:=\:\mathrm{19} \\ $$$$\mathrm{or}\:{a}\:+\:\mathrm{6}{r}\:=\:\mathrm{13}\:\:\:\left(\mathrm{3}\right) \\ $$$$\left(\mathrm{2}\right)\:\mathrm{in}\:\left(\mathrm{3}\right)\:\Rightarrow\:\:{a}\:+\:\mathrm{6}\left(\frac{\mathrm{6}}{{a}}\right)\:=\:\mathrm{13} \\ $$$$\Rightarrow\:{a}^{\mathrm{2}} \:−\mathrm{13}{a}\:+\:\mathrm{36}\:=\:\mathrm{0}\: \\ $$$$\:\:\:\:\:\:\Leftrightarrow\:{a}\:=\:\mathrm{4}\:\mathrm{or}\:{a}\:=\:\mathrm{9} \\ $$$$\Rightarrow\:{r}\:=\:\frac{\mathrm{3}}{\mathrm{2}}\:\mathrm{or}\:{r}\:=\:\frac{\mathrm{2}}{\mathrm{3}} \\ $$$$\mathrm{now}\:\mathrm{you}\:\mathrm{can}\:\mathrm{write}\:\mathrm{out}\:\mathrm{two}\:\mathrm{sequences}. \\ $$