Question Number 140462 by ZiYangLee last updated on 08/May/21
Commented by mr W last updated on 08/May/21
$${T}_{{n}} ={ar}^{{n}−\mathrm{1}} \\ $$$$\frac{{T}_{{n}+\mathrm{4}} −{T}_{{n}+\mathrm{2}} }{{T}_{{n}+\mathrm{1}} }=\frac{{ar}^{{n}+\mathrm{3}} −{ar}^{{n}+\mathrm{1}} }{{ar}^{{n}} }={r}^{\mathrm{3}} −{r} \\ $$$$=\left({r}−\mathrm{1}\right){r}\left({r}+\mathrm{1}\right) \\ $$$${one}\:{of}\:{r}\:{and}\:{r}+\mathrm{1}\:{is}\:{even},\:{so}\:\left({r}−\mathrm{1}\right){r}\left({r}+\mathrm{1}\right) \\ $$$${is}\:{always}\:{even}. \\ $$