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Question Number 144264 by loveineq last updated on 24/Jun/21
Let a,b > 0 and 2a+b = 3. Prove the followings:            (1)                (2/n)a(b+4)+3b^(1/n)  ≤ ((10+3n)/n), ∀n∈N^+ ≥1.  (2)                2na(b+4)+3b^n  ≥ 10n+3, ∀n∈N^+ ≥2.
$$\mathrm{Let}\:{a},{b}\:>\:\mathrm{0}\:\mathrm{and}\:\mathrm{2}{a}+{b}\:=\:\mathrm{3}.\:\mathrm{Prove}\:\mathrm{the}\:\mathrm{followings}:\:\:\:\:\:\:\:\:\:\: \\ $$$$\left(\mathrm{1}\right)\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\frac{\mathrm{2}}{{n}}{a}\left({b}+\mathrm{4}\right)+\mathrm{3}{b}^{\frac{\mathrm{1}}{{n}}} \:\leqslant\:\frac{\mathrm{10}+\mathrm{3}{n}}{{n}},\:\forall{n}\in\mathbb{N}^{+} \geqslant\mathrm{1}. \\ $$$$\left(\mathrm{2}\right)\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{2}{na}\left({b}+\mathrm{4}\right)+\mathrm{3}{b}^{{n}} \:\geqslant\:\mathrm{10}{n}+\mathrm{3},\:\forall{n}\in\mathbb{N}^{+} \geqslant\mathrm{2}. \\ $$$$ \\ $$

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