Menu Close

Given-a-b-c-N-prove-that-1-a-1-2a-1-b-1-2b-1-c-1-2c-2-




Question Number 7782 by Chantria last updated on 15/Sep/16
 Given a,b,c ∈N ; prove that   ((1+a)/(1+2a)) + ((1+b)/(1+2b)) + ((1+c)/(1+2c)) ≤ 2
$$\:{Given}\:{a},{b},{c}\:\in{N}\:;\:{prove}\:{that} \\ $$$$\:\frac{\mathrm{1}+{a}}{\mathrm{1}+\mathrm{2}{a}}\:+\:\frac{\mathrm{1}+{b}}{\mathrm{1}+\mathrm{2}{b}}\:+\:\frac{\mathrm{1}+{c}}{\mathrm{1}+\mathrm{2}{c}}\:\leqslant\:\mathrm{2} \\ $$
Commented by sou1618 last updated on 15/Sep/16
Let f(n)=((1+n)/(1+2n))  (n≥1)    f ′(n)=((1×(1+2n)−(1+n)×2)/((1+2n)^2 ))  f ′(n)=((−1)/((1+2n)^2 ))<0  so  f(n)≤f(1)      (n≥1)  f(1)=(2/3)    ((1+a)/(1+2a))+((1+b)/(1+2b))+((1+c)/(1+2c))≤3f(n)≤2
$${Let}\:{f}\left({n}\right)=\frac{\mathrm{1}+{n}}{\mathrm{1}+\mathrm{2}{n}}\:\:\left({n}\geqslant\mathrm{1}\right) \\ $$$$ \\ $$$${f}\:'\left({n}\right)=\frac{\mathrm{1}×\left(\mathrm{1}+\mathrm{2}{n}\right)−\left(\mathrm{1}+{n}\right)×\mathrm{2}}{\left(\mathrm{1}+\mathrm{2}{n}\right)^{\mathrm{2}} } \\ $$$${f}\:'\left({n}\right)=\frac{−\mathrm{1}}{\left(\mathrm{1}+\mathrm{2}{n}\right)^{\mathrm{2}} }<\mathrm{0} \\ $$$${so} \\ $$$${f}\left({n}\right)\leqslant{f}\left(\mathrm{1}\right)\:\:\:\:\:\:\left({n}\geqslant\mathrm{1}\right) \\ $$$${f}\left(\mathrm{1}\right)=\frac{\mathrm{2}}{\mathrm{3}} \\ $$$$ \\ $$$$\frac{\mathrm{1}+{a}}{\mathrm{1}+\mathrm{2}{a}}+\frac{\mathrm{1}+{b}}{\mathrm{1}+\mathrm{2}{b}}+\frac{\mathrm{1}+{c}}{\mathrm{1}+\mathrm{2}{c}}\leqslant\mathrm{3}{f}\left({n}\right)\leqslant\mathrm{2} \\ $$$$ \\ $$

Leave a Reply

Your email address will not be published. Required fields are marked *