Mathematics - Sequence & Series Question with Solution | TestHub

Take the seven positive numbers $a,a^2,a^2,a^3,a^{-1},a^{-3},a^{-4}$ and apply A.M. $\ge$ G.M. $\frac{a+2a^2+a^3+a^{-1}+a^{-3}+a^{-4}}{7}\ge (1{)}^{1/7}=1,\text{ so the sum is at least }7\text{ (equality at }a=1\text{ ). }$ Minimum value $=7+15=22$.