Hãy nhập câu hỏi của bạn vào đây, nếu là tài khoản VIP, bạn sẽ được ưu tiên trả lời.
Vì \(n\in Z^+\)nên\(n\left(n+1\right)\left(n+2\right)>n^3\Rightarrow\sqrt[3]{n\left(n+1\right)\left(n+2\right)}>n\)
\(\Rightarrow\sqrt[3]{n\left(n+1\right)\left(n+2\right)+\sqrt[3]{n\left(n+1\right)\left(n+2\right)}+...+\sqrt[3]{n\left(n+1\right)\left(n+2\right)}}>n\)(1)
Lại có:\(n^2+2n+1>n^2+2n\Rightarrow\left(n+1\right)^2>n\left(n+2\right)\Rightarrow\left(n+1\right)^3>n\left(n+1\right)\left(n+2\right)\)
\(\Rightarrow n+1>\sqrt[3]{n\left(n+1\right)\left(n+2\right)}\\ \Rightarrow\sqrt[3]{n^3+3n^2+3n+1}>\sqrt[3]{n^3+3n^2+2n}\)
\(\Rightarrow\sqrt[3]{n^3+3n^2+2n+n+1}>\sqrt[3]{n^3+3n^2+2n+\sqrt[3]{n\left(n+1\right)\left(n+2\right)}}\)
\(\Rightarrow\sqrt[3]{\left(n+1\right)^3}>\sqrt[3]{n\left(n+1\right)\left(n+2\right)+\sqrt[3]{n\left(n+1\right)\left(n+2\right)}}\)
Tương tự \(\Rightarrow n+1>\sqrt[3]{n\left(n+1\right)\left(n+2\right)+\sqrt[3]{n\left(n+1\right)\left(n+2\right)}+...+\sqrt[3]{n\left(n+1\right)\left(n+2\right)}}\)(2)
Từ (1) và (2) suy ra:
\(n< \sqrt[3]{n\left(n+1\right)\left(n+2\right)+\sqrt[3]{n\left(n+1\right)\left(n+2\right)}+...+\sqrt[3]{n\left(n+1\right)\left(n+2\right)}}< n+1\)
\(n\in Z^+\)nên n2 < n2 + 2n < n2 + 2n + 1 <=> n2 < n(n + 2) < (n + 1)2 => n3 < n(n + 1)(n + 2) < (n + 1)3
=>\(n< \sqrt[3]{n\left(n+1\right)\left(n+2\right)}< n+1\)
=>\(n< \sqrt[3]{n\left(n+1\right)\left(n+2\right)}< \sqrt[3]{n\left(n+1\right)\left(n+2\right)+n}\)\(< \sqrt[3]{n\left(n+1\right)\left(n+2\right)+\sqrt[3]{n\left(n+1\right)\left(n+2\right)}}< \sqrt[3]{n\left(n+1\right)\left(n+2\right)+n+1}\)\(=\sqrt[3]{\left(n+1\right)\left(n^2+2n+1\right)}=\sqrt[3]{\left(n+1\right)\left(n+1\right)^2}=n+1\)
=>\(n< \sqrt[3]{n\left(n+1\right)\left(n+2\right)+n}\)
\(< \sqrt[3]{n\left(n+1\right)\left(n+2\right)+\sqrt[3]{n\left(n+1\right)\left(n+2\right)+\sqrt[3]{n\left(n+1\right)\left(n+2\right)}}}< n+1\)
Tiếp tục như vậy,ta có đpcm.
Ta có từ n3 + 1 đến (n + 1)3 - 1 có
(n + 1)3 - 1 - n3 - 1 + 1 = 3n2 + 3n số có phần nguyên bằng n
Áp dụng vào cái ban đầu ta có
\(=\frac{3.1^2+3.1}{1}+\frac{3.2^2+3.2}{2}+...+\frac{3.2011^2+3.2011}{2011}\)
= 3.1 + 3 + 3.2 + 3 + ...+ 3.2011 + 3
= 3.2011 + 3(1 + 2 +...+ 2011)
= 6075231
\(f\left(n\right)=\dfrac{2n-1+2n+1+\sqrt{\left(2n+1\right)\left(2n+1\right)}}{\sqrt{2n+1}+\sqrt{2n-1}}\\ f\left(n\right)=\dfrac{\left(\sqrt{2n+1}-\sqrt{2n-1}\right)\left(2n-1+2n+1+\sqrt{\left(2n+1\right)\left(2n+1\right)}\right)}{2n+1-2n+1}\\ f\left(n\right)=\dfrac{\left(\sqrt{2n+1}\right)^3-\left(\sqrt{2n+1}\right)^3}{2}=\dfrac{\left(2n+1\right)\sqrt{2n+1}-\left(2n-1\right)\sqrt{2n+1}}{2}\)
\(\Leftrightarrow f\left(1\right)+f\left(2\right)+...+f\left(40\right)=\dfrac{3\sqrt{3}-1\sqrt{1}+5\sqrt{5}-3\sqrt{3}+...+81\sqrt{81}-79\sqrt{79}}{2}\\ =\dfrac{81\sqrt{81}-1\sqrt{1}}{2}=\dfrac{9^3-1}{2}=364\)
Ta co:
\(\frac{1}{\left(2n+1\right)\left(\sqrt{n}+\sqrt{n+1}\right)}=\frac{\sqrt{n+1}-\sqrt{n}}{n+1+n}< \frac{\sqrt{n+1}-\sqrt{n}}{2\sqrt{n+1}.\sqrt{n}}=\frac{1}{2}\left(\frac{1}{\sqrt{n}}-\frac{1}{\sqrt{n+1}}\right)\)
Ap vào bài toan được
\(S_n=\frac{1}{3\left(1+\sqrt{2}\right)}+\frac{1}{5\left(\sqrt{2}+\sqrt{3}\right)}+...+\frac{1}{\left(2n+1\right)\left(\sqrt{n}+\sqrt{n+1}\right)}\)
\(< \frac{1}{2}\left(\frac{1}{1}-\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}}-\frac{1}{\sqrt{3}}+...+\frac{1}{\sqrt{n}}-\frac{1}{\sqrt{n+1}}\right)\)
\(=\frac{1}{2}\left(1-\frac{1}{\sqrt{n+1}}\right)< \frac{1}{2}\)
Ta có \(n\left(n+1\right)\left(n+2\right)\left(n+3\right)=n\left(n+3\right)\left(n+1\right)\left(n+2\right)\)
\(=\left(n^2+3n\right)\left(n^2+3n+2\right)\)
Đặt \(n^2+3n=a\in N\Rightarrow\left(n^2+3n\right)\left(n^2+3n+2\right)=a\left(a+2\right)\)
\(=a^2+2a\)
Mà \(a^2\le a^2+2a< a^2+2a+1\Rightarrow a^2\le a^2+2a< \left(a+1\right)^2\)
\(\Rightarrow a\le\sqrt{a^2+2a}< a+1\Rightarrow a\le\left[\sqrt{a^2+2a}\right]< a+1\)
\(\Rightarrow\left[\sqrt{a^2+2a}\right]=a\)
\(\Rightarrow\left[n\left(n+1\right)\left(n+2\right)\left(n+3\right)\right]=n^2+3n=n\left(n+3\right)\)
Vậy:
\(\sum\sqrt{n\left(n+1\right)\left(n+2\right)\left(n+3\right)}=\sum n\left(n+3\right)=\dfrac{n\left(n+1\right)\left(n+5\right)}{3}\)