Question Number 65901 by byaw last updated on 05/Aug/19
Answered by MJS last updated on 05/Aug/19
$${a}_{\mathrm{0}} \\ $$$${a}_{\mathrm{3}} ={a}_{\mathrm{0}} +\mathrm{2}\alpha \\ $$$${a}_{\mathrm{8}} ={a}_{\mathrm{0}} +\mathrm{7}\alpha \\ $$$${g}_{\mathrm{0}} ={a}_{\mathrm{0}} \\ $$$${g}_{\mathrm{1}} =\gamma{g}_{\mathrm{0}} ={a}_{\mathrm{3}} \\ $$$${g}_{\mathrm{2}} =\gamma{g}_{\mathrm{1}} ={a}_{\mathrm{8}} \\ $$$$\gamma=\frac{{a}_{\mathrm{3}} }{{a}_{\mathrm{0}} }=\frac{{a}_{\mathrm{8}} }{{a}_{\mathrm{3}} } \\ $$$$\Rightarrow\:{a}_{\mathrm{3}} ^{\mathrm{2}} ={a}_{\mathrm{0}} {a}_{\mathrm{8}} \\ $$$$\left({a}_{\mathrm{0}} +\mathrm{2}\alpha\right)^{\mathrm{2}} ={a}_{\mathrm{0}} \left({a}_{\mathrm{0}} +\mathrm{7}\alpha\right) \\ $$$$\mathrm{4}{a}_{\mathrm{0}} +\mathrm{4}\alpha=\mathrm{7}{a}_{\mathrm{0}} \\ $$$$\alpha=\frac{\mathrm{3}}{\mathrm{4}}{a}_{\mathrm{0}} \\ $$$${g}_{\mathrm{0}} ={a}_{\mathrm{0}} \\ $$$${g}_{\mathrm{1}} =\frac{\mathrm{5}}{\mathrm{2}}{a}_{\mathrm{0}} \\ $$$${g}_{\mathrm{2}} =\frac{\mathrm{25}}{\mathrm{4}}{a}_{\mathrm{0}} \\ $$$$\Rightarrow\:\gamma=\frac{\mathrm{5}}{\mathrm{2}} \\ $$$${g}_{\mathrm{0}} +{g}_{\mathrm{1}} +{g}_{\mathrm{2}} =\frac{\mathrm{39}}{\mathrm{4}}{a}_{\mathrm{0}} =\mathrm{28}\:\Rightarrow\:{a}_{\mathrm{0}} =\frac{\mathrm{112}}{\mathrm{39}}={g}_{\mathrm{0}} \\ $$$$\Rightarrow\:\alpha=\frac{\mathrm{28}}{\mathrm{13}} \\ $$