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Q.
If $x, y, z$ and $w$ are positive integers such that $x+2 y+3 z$ $+4 w=50,$ then maximum value of $\left(\frac{x^{2} y^{4} z^{3} w}{16}\right)^{1 / 10}$ is
Sequences and Series
Solution:
We have $x+2 y+3 z+4 w=50$
Using the fact A.M. $\geq$ G.M., we get
$\frac{2\left(\frac{x}{2}\right)+4\left(\frac{y}{2}\right)+3\left(\frac{z}{1}\right)+1\left(\frac{4 w}{1}\right)}{2+4+3+1}=\frac{50}{10} \geq\left[\left(\frac{x}{2}\right)^{2}\left(\frac{y}{2}\right)^{4}(z)^{3}(4 w)\right]^{1 / 10}$
$\Rightarrow 5 \geq\left[\left(\frac{x^{2}}{2^{2}}\right)\left(\frac{y^{4}}{2^{4}}\right)(z)^{3}\left(2^{2} w\right)\right]^{1 / 10}$
$\Rightarrow 5 \geq\left(\frac{x^{2} y^{4} z^{3} w}{16}\right)^{1 / 10}$