Question

1. tính nhanh

(\(\frac{1}{3}\)+\(\frac{12}{67}\)+\(\frac{13}{41}\)) - (\(\frac{79}{67}\)-\(\frac{28}{41}\))

2. tìm x

\(\frac{1}{2}\). x + \(\frac{2}{1.2.3}\)+ \(\frac{2}{2.3.4}\)+............+\(\frac{2}{98.99.100}\)

 

 

Answer 1

1)

\(=\frac{1}{3}+\frac{12}{67}+\frac{13}{41}-\frac{79}{67}+\frac{28}{41}\)

\(=\frac{1}{3}+\left(\frac{12}{67}-\frac{79}{67}\right)+\left(\frac{13}{41}+\frac{28}{41}\right)=\frac{1}{3}+\left(-1\right)+1=\frac{1}{3}\)

Answer 2

Sửa đề chút nha

\(\frac{x}{2}=\frac{1}{1.2.3}+....+\frac{1}{98.99.100}\)

Ta có công thức tổng quát  \(\frac{1}{a\left(a+1\right)\left(a+2\right)}=\frac{1}{2}\left(\frac{1}{a\left(a+1\right)}-\frac{1}{\left(a+1\right)\left(a+2\right)}\right)\)

\(\Rightarrow\frac{2}{a\left(a+1\right)\left(a+2\right)}=\frac{1}{a\left(a+1\right)}-\frac{1}{\left(a+1\right)\left(a+2\right)}\)

Áp dụng vào tổng ta có

\(\frac{x}{2}=\frac{1}{1.2}-\frac{1}{2.3}+\frac{1}{2.3}-\frac{1}{3.4}+....+\frac{1}{98.99}-\frac{1}{99.100}=\frac{1}{2}-\frac{1}{99.100}=\frac{4949}{9900}\)

\(\Rightarrow x=\frac{4949}{4950}\)