1. \(\frac{2016}{2017}\)+\(\frac{2017}{2018}\)>1

2. A>B

\(B=\sqrt{1+2017^2+\frac{2017^2}{2018^2}}+\frac{2017}{2018}\)

\(B=\sqrt{\left(1+2.2017+2017^2\right)-2.2017+\frac{2017^2}{2018^2}}+\frac{2017}{2018}\)

\(B=\sqrt{\left(1+2017\right)^2-2.2017+\frac{2017^2}{2018^2}}+\frac{2017}{2018}\)

\(B=\sqrt{2018^2-2.2017+\frac{2017^2}{2018^2}}+\frac{2017}{2018}\)

\(B=\sqrt{\left(2018-\frac{2017}{2018}\right)^2}+\frac{2017}{2018}\)

Mà  \(\frac{2017}{2018}< 1\Rightarrow2018-\frac{2017}{2018}>0\)

\(\Rightarrow B=2018-\frac{2017}{2018}+\frac{2017}{2018}\)

\(B=2018\)

Vậy bt B có giá trị nguyên 

Cảm ơn bạn mk vừa đăng lên thì đã thấy luôn cách giải 😂

a) \(A=\frac{2+2^2+...+2^{2017}}{1-2^{2017}}\)

Đặt \(B=2+2^2+...+2^{2017}\)

\(\Rightarrow2B=2^2+2^3+...+2^{2018}\)

\(\Rightarrow2B-B=\left(2^2+2^3+...+2^{2018}\right)-\left(2+...+2^{2017}\right)\)

\(\Rightarrow B=2^{2018}-2\)

\(\Rightarrow A=\frac{2^{2018}-2}{1-2^{2017}}\)

\(\Rightarrow A=\frac{-2.\left(1-2^{2017}\right)}{1-2^{2017}}\)

\(\Rightarrow A=-2\)

b)Đề phải là CM: \(A< \frac{2017}{2016^2}\)

 \(A=\frac{1}{2017}+\frac{2}{2017^2}+...+\frac{22017}{2017^{2017}}+\frac{2018}{2017^{2018}}\)

\(\Rightarrow2017A=1+\frac{2}{2017}+...+\frac{22017}{2017^{2016}}+\frac{2018}{2017^{2017}}\)

\(\Rightarrow2017A-A=\left(1+...+\frac{2018}{2017^{2017}}\right)-\left(\frac{1}{2017}+...+\frac{2017}{2017^{2017}}+\frac{2018}{2017^{2018}}\right)\)

\(\Rightarrow2016A=1+\frac{1}{2017}+\frac{1}{2017^2}+...+\frac{1}{2017^{2017}}-\frac{2018}{2017^{2018}}\)

Đặt \(\Rightarrow S=1+\frac{1}{2017}+\frac{1}{2017^2}+...+\frac{1}{2017^{2017}}\)

\(\Rightarrow2017S=2017+1+\frac{1}{2017}+...+\frac{1}{2017^{2016}}\)

\(\Rightarrow2017S-S=\left(2017+1+...+\frac{1}{2017^{2016}}\right)-\left(1+...+\frac{1}{2017^{2017}}\right)\)

\(\Rightarrow2016S=2017-\frac{1}{2017^{2017}}< 2017\)

\(\Rightarrow2016S< 2017\)

\(\Rightarrow S< \frac{2017}{2016}\)

\(\Rightarrow2016A< \frac{2017}{2016}\)

\(\Rightarrow A< \frac{2017}{2016^2}\left(đpcm\right)\)

Lời giải:

\(\sqrt{1+2017^2+\frac{2017^2}{2018^2}}+\frac{2017}{2018}\)

\(=\sqrt{(1+2017)^2-2.2017+\frac{2017^2}{2018^2}}+\frac{2017}{2018}\)

\(\sqrt{2018^2-2.2018.\frac{2017}{2018}+\frac{2017^2}{2018^2}}+\frac{2017}{2018}\)

\(=\sqrt{(2018-\frac{2017}{2018})^2}+\frac{2017}{2018}=2018-\frac{2017}{2018}+\frac{2017}{2018}=2018\)

Câu b đề sai nha, bây giờ đặt \(a=\sqrt{2017},b=\sqrt{2018}\)

Ta có ^{\(\frac{a^2}{b}+\frac{b^2}{a}< a+b\Leftrightarrow ab\left(\frac{a^2}{b}+\frac{b^2}{a}\right)< ab\left(a+b\right)\)}

\(\Leftrightarrow a^3+b^3< ab\left(a+b\right)\)(1)

Mà \(ab\left(a+b\right)\le\left(a^2-ab+b^2\right)\left(a+b\right)=a^3+b^3\)(2)

Từ (1), (2) => Sai

a) Ta có:

\(\frac{1}{\left(k+1\right)\sqrt{k}}=\frac{k+1-k}{\left(k+1\right)\sqrt{k}}=\frac{\left(\sqrt{k+1}+\sqrt{k}\right)\left(\sqrt{k+1}-\sqrt{k}\right)}{\left(k+1\right)\sqrt{k}}\)\(< \frac{2\sqrt{k+1}\left(\sqrt{k+1}-\sqrt{k}\right)}{\left(k+1\right)\sqrt{k}}=\frac{2\left(\sqrt{k+1}-\sqrt{k}\right)}{\sqrt{k+1}\sqrt{k}}=\frac{2}{\sqrt{k}}-\frac{2}{\sqrt{k+1}}\)

Cho k=1,2,....,n rồi cộng từng vế ta có:

\(\frac{1}{2}+\frac{1}{3\sqrt{2}}+\frac{1}{4\sqrt{3}}+....+\frac{1}{\left(n+1\right)\sqrt{n}}< \left(\frac{2}{\sqrt{1}}-\frac{2}{\sqrt{2}}\right)+\left(\frac{2}{\sqrt{2}}-\frac{2}{\sqrt{3}}\right)\)\(+\left(\frac{2}{\sqrt{3}}-\frac{2}{\sqrt{4}}\right)+....+\left(\frac{2}{\sqrt{n}}-\frac{2}{\sqrt{n+1}}\right)=2-\frac{2}{\sqrt{n-1}}< 2\)

\(B=\sqrt{1+2017^2+\frac{2017^2}{2018^2}}+\frac{2017}{2018}\)

Đặt B = 2017 => B + 1 = 2018

Khi B bằng: 

\(B=\sqrt{1+B^2+\frac{B}{\left(B+1\right)^2}}+\frac{B}{B+1}\)

\(B=\sqrt{\frac{\left(B+1\right)^2+B^2\left(B+1\right)^2+B^2}{\left(B+1\right)^2}}+\frac{B}{B+1}\)

\(B=\sqrt{\frac{B^2\left(B+1\right)^2+2B\left(B+1\right)^2+B^2}{\left(B+1\right)^2}}+\frac{B}{B+1}\)

\(B=\sqrt{\frac{\left[B\left(B+1\right)+1\right]^2}{\left(B+1\right)^2}}+\frac{B}{B+1}\)

\(B=\frac{B^2+B+1}{B+1}+\frac{B}{B+1}\left(\text{vi}:a>0\right)\)

\(B=\frac{B^2+2B+1}{B+1}\)

\(B=\frac{\left(B+1\right)^2}{B+1}\)

\(B=B+1\left(\text{vi}:a>0\Rightarrow B+1>0\right)\)

\(B=2017+1\left(\text{vi}:B=2017\right)\)

\(\Rightarrow B=2018\)

Đặt \(2017=a\)

=>\(2018=a+1\)

Với mọi \(a\in N\) có:\(\sqrt{1+a^2+\frac{a^2}{\left(a+1\right)^2}}=\sqrt{\frac{\left(a+1\right)^2+a^2\left(a+1\right)^2+a^2}{\left(a+1\right)^2}}=\sqrt{\frac{a^2+2a+1+a^2\left(a^2+2a+1\right)+a^2}{\left(a+1\right)^2}}=\sqrt{\frac{2a^2+2a+1+a^4+2a^3+a^2}{\left(a+1\right)^2}}=\sqrt{\frac{\left(a^4+2a^2+1\right)+2a\left(a^2+1\right)+a^2}{\left(a+1\right)^2}}\)

=\(\sqrt{\frac{\left(a^2+1\right)^2+2a\left(a^2+1\right)+a^2}{\left(a+1\right)^2}}=\sqrt{\frac{\left(a^2+a+1\right)}{\left(a+1\right)^2}}=\left|\frac{a^2+a+1}{a+1}\right|\)(do \(a\ge0\))

=\(\frac{a\left(a+1\right)+1}{a+1}=a+\frac{1}{a+1}\)

=> \(\sqrt{1+a^2+\frac{a^2}{\left(a+1\right)^2}}=a+\frac{1}{a+1}\)

Thay a=2017 có:

\(\sqrt{1+2017^2+\left(\frac{2017}{2018}\right)^2}=2017+\frac{1}{2017+1}=2017+\frac{1}{2018}\)

=>\(\sqrt{1+22017^2+\left(\frac{2017}{2018}\right)^2}+\frac{2017}{2018}=2017+\frac{1}{2018}+\frac{2017}{2018}\)

<=> M=2017+1=2018

Vậy M=2018

Vũ Minh Tuấn Lê Thị Thục Hiền @No choice teen

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