Mathematics - Sequence & Series Question with Solution | TestHub

Take the ten positive numbers $\frac{a}{2},\frac{a}{2},\frac{b}{2},\frac{b}{2},\frac{b}{2},\frac{b}{2},c,c,c,4d$ and apply A.M. $\ge$ G.M. $\frac{a+2b+3c+4d}{10}\ge {\left(\frac{a^2}{4}\cdot \frac{b^4}{16}\cdot c^3\cdot 4d\right)}^{1/10}={\left(\frac{a^2{b}^4{c}^{3}d}{16}\right)}^{1/10}.$

So ${\left(\frac{a^2{b}^4{c}^{3}d}{16}\right)}^{1/10}\le \frac{50}{10}=5$. Maximum value $=5$.