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Question Number 7782 by Chantria last updated on 15/Sep/16

$$Givena,b,c\in N;provethat$$$$\frac{1+a}{1+2a}+\frac{1+b}{1+2b}+\frac{1+c}{1+2c}⩽2$$

Commented by sou1618 last updated on 15/Sep/16

$$Letf\left(n\right)=\frac{1+n}{1+2n}(n⩾1)$$$$$$$$f{}^{\prime }\left(n\right)=\frac{1\times (1+2n)-(1+n)\times 2}{{(1+2n)}^2}$$$$f{}^{\prime }\left(n\right)=\frac{-1}{{(1+2n)}^2}<0$$$$so$$$$f\left(n\right)⩽f\left(1\right)(n⩾1)$$$$f\left(1\right)=\frac{2}{3}$$$$$$$$\frac{1+a}{1+2a}+\frac{1+b}{1+2b}+\frac{1+c}{1+2c}⩽3f\left(n\right)⩽2$$$$$$

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