Lời giải:

Xét công thức tổng quát:

$1+2+3+...+n=\frac{n(n+1)}{2}$

$\Rightarrow 1-\frac{1}{1+2+3+...+n}=1-\frac{2}{n(n+1)}=\frac{(n-1)(n+2)}{n(n+1)}$

Thay $n=2,3,...,2006$ ta thu được:

\(A=\frac{1.4}{2.3}.\frac{2.5}{3.4}.\frac{3.6}{4.5}....\frac{2005.2008}{2006.2007}\)

\(=\frac{(1.2.3...2005)(4.5.6...2008)}{(2.3.4...2006)(3.4.5...2007)}=\frac{1}{2006}.\frac{2008}{3}=\frac{1004}{3009}\)

Lời giải:

Xét công thức tổng quát:

$1+2+3+...+n=\frac{n(n+1)}{2}$

$\Rightarrow 1-\frac{1}{1+2+3+...+n}=1-\frac{2}{n(n+1)}=\frac{(n-1)(n+2)}{n(n+1)}$

Thay $n=2,3,...,2006$ ta thu được:

\(A=\frac{1.4}{2.3}.\frac{2.5}{3.4}.\frac{3.6}{4.5}....\frac{2005.2008}{2006.2007}\)

\(=\frac{(1.2.3...2005)(4.5.6...2008)}{(2.3.4...2006)(3.4.5...2007)}=\frac{1}{2006}.\frac{2008}{3}=\frac{1004}{3009}\)

\(x\)là dấu nhân hả bạn? Nếu vậy thì mk làm cho nhé

\(A=\left(1-\frac{1}{2}\right)\cdot\left(1-\frac{1}{3}\right)\cdot\left(1-\frac{1}{4}\right)\cdot....\cdot\left(1-\frac{1}{20}\right)\)

\(A=\frac{1}{2}\cdot\frac{2}{3}\cdot\frac{3}{4}\cdot.......\cdot\frac{17}{18}\cdot\frac{18}{19}\cdot\frac{19}{20}=\frac{1}{20}\)

Vậy \(A=\frac{1}{20}\)

\(B=1\frac{1}{2}\cdot1\frac{1}{3}\cdot1\frac{1}{4}\cdot........\cdot1\frac{1}{2005}\cdot1\frac{1}{2006}\cdot1\frac{1}{2007}\)

\(B=\frac{3}{2}\cdot\frac{4}{3}\cdot\frac{5}{4}\cdot......\cdot\frac{2006}{2005}\cdot\frac{2007}{2006}\cdot\frac{2008}{2007}=\frac{2008}{2}=1004\)

Vậy \(B=1004\)

DẤU CHẤM LÀ DẤU NHÂN

a, 

\(=\frac{1}{2}.\frac{2}{3}.\frac{3}{4}....\frac{19}{20}=\frac{1}{20}\)

b, \(1\frac{1}{2}.1\frac{1}{3}....1\frac{1}{2017}=\frac{3}{2}.\frac{4}{3}....\frac{2018}{2017}=\frac{2018}{2}=1009\)

Ta thấy: \(1-\frac{1}{1+2+3+...+n}=1-\frac{2}{n\left(n+1\right)}=\frac{n^2+n-2}{n\left(n+1\right)}=\frac{\left(n-1\right)\left(n+2\right)}{n\left(n+1\right)}\)

\(\Rightarrow A=\left(1-\frac{1}{1+2}\right)\left(1-\frac{1}{1+2+3}\right)\left(1-\frac{1}{1+2+3+4}\right)...\left(1-\frac{1}{1+2+3+...+2006}\right)\)

\(=\left(\frac{\left(2-1\right)\left(2+2\right)}{2\left(2+1\right)}\right)\left(\frac{\left(3-1\right)\left(3+2\right)}{3\left(3+1\right)}\right)\left(\frac{\left(4-1\right)\left(4+2\right)}{4\left(4+1\right)}\right)...\left(\frac{\left(2006-1\right)\left(2006+2\right)}{2006\left(2006+1\right)}\right)\)

\(=\frac{1.4}{2.3}.\frac{2.5}{3.4}.\frac{3.6}{4.5}...\frac{2005.2008}{2006.2007}=\frac{\left(1.2.3...2005\right)\left(4.5.6...2008\right)}{\left(2.3.4...2006\right)\left(3.4.5...2007\right)}\)

\(=\frac{1.2008}{2006.3}=\frac{1004}{1003.3}=\frac{1004}{3009}\)

Vậy \(A=\frac{1004}{3009}\)

\(VT=\frac{2\left(\sqrt{2}-\sqrt{1}\right)}{3.\left(2-1\right)}+\frac{2\left(\sqrt{3}-\sqrt{2}\right)}{5\left(3-2\right)}+...+\frac{2\left(\sqrt{2006}-\sqrt{2005}\right)}{4011\left(2006-2005\right)}\)

\(VT=\frac{2\left(\sqrt{2}-\sqrt{1}\right)}{3}+\frac{2\left(\sqrt{3}-\sqrt{2}\right)}{5}+...+\frac{2\left(\sqrt{2006}-\sqrt{2005}\right)}{4011}\)

Nhận xét: (a-b)² \(\ge\) 0 => a² + b²  \(\ge\) 2ab

Áp dụng ta có: \(3=\left(\sqrt{2}\right)^2+\left(\sqrt{1}\right)^2\ge2.\sqrt{2}.\sqrt{1}\)

\(5=\left(\sqrt{3}\right)^2+\left(\sqrt{2}\right)^2\ge2.\sqrt{3}.\sqrt{2}\)

...

\(4011=\left(\sqrt{2006}\right)^2+\left(\sqrt{2005}\right)^2\ge2.\sqrt{2006}.\sqrt{2005}\)

=> \(VT

a) \(\frac{\left(-1\right)}{4}^2+\frac{3}{8}.\left(\frac{-1}{6}\right)-\frac{3}{16}:\left(\frac{-1}{2}\right)=\left(\frac{-1}{4}\right)^2+\left(\frac{-3}{68}\right)-\left(\frac{-3}{8}\right)=\left(\frac{1}{16}\right)+\left(\frac{-3}{68}\right)-\left(\frac{-3}{8}\right)=\frac{5}{272}-\left(\frac{-3}{8}\right)=\frac{107}{272}\)

\(A=(1-\frac{1}{1+2})(1-\frac{1}{1+2+3})(1-\frac{1}{1+2+3+4})...(1-\frac{1}{1+2+3+...+2006})\)

\(A=(1-\frac{1}{3})(1-\frac{1}{6})(1-\frac{1}{10})...(1-\frac{1}{2013021})\)

\(A=\frac{2}{3}\cdot\frac{5}{6}\cdot\frac{9}{10}....\frac{2013020}{2013021}\)

Sorry bạn máy tính mình có chút vấn đề để mk làm tiếp :

\(A=\frac{4}{6}\cdot\frac{10}{12}\cdot\frac{18}{20}....\cdot\frac{4026040}{4026042}\)

\(A=\frac{1\cdot4}{2\cdot3}\cdot\frac{2\cdot5}{3\cdot4}\cdot\frac{3\cdot6}{4\cdot5}\cdot...\cdot\frac{2005\cdot2008}{2006\cdot2007}\)

\(A=\frac{1\cdot2\cdot3\cdot...\cdot2005}{2\cdot3\cdot4\cdot...\cdot2006}\cdot\frac{4\cdot5\cdot6\cdot...\cdot2008}{3\cdot4\cdot5\cdot...\cdot2007}\)

\(A=\frac{1}{2006}\cdot\frac{2008}{3}=\frac{1004}{3009}\)

P/S : Hoq chắc :>

dsf