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![](https://rs.olm.vn/images/avt/0.png?1311)
B= 1/ 1.2.3 + 1/ 2.3 4 + 1/ 3.4.5 + .... + 1/ 48.49.50
Mà ta có:
1/ 1.2 - 1/ 2.3 = 2/ 1.2.3
1/ 2.3 - 1/3.4 = 2/ 2.3.4
Từ đó=> B = 1/2 . ( 2/ 1.2.3 + 2/ 2,3.4 + ... + 2/ 18. 19. 20 )
= 1/2 .( 1/ 1.2 – 1/ 2.3 + 1/ 2.3 - .....- 1/19.20)
= 1/2. ( 1/ 1.2 – 1/ 19.20 ) = 1/ 2 . 189/380 = 189/760
![](https://rs.olm.vn/images/avt/0.png?1311)
Ta có ; \(1\dfrac{1}{10}.1\dfrac{1}{11}.....1\dfrac{1}{2011}.1\dfrac{1}{2012}\)
\(=\dfrac{11}{10}.\dfrac{12}{11}.....\dfrac{2012}{2011}.\dfrac{2013}{2012}\)
\(=\dfrac{2013}{10}=201,3\)
1 \(\dfrac{1}{10}\)x 1\(\dfrac{1}{11}\)x 1\(\dfrac{1}{12}\) x...x 1\(\dfrac{1}{2011}\)x 1\(\dfrac{1}{2012}\)
=\(\dfrac{11}{10}\)x \(\dfrac{12}{11}\)x \(\dfrac{13}{12}\)x \(\dfrac{2012}{2011}\)x \(\dfrac{2013}{2012}\)
![](https://rs.olm.vn/images/avt/0.png?1311)
Đặt A = 1/2+1/4+1/8+1/18+1/32+1/64+1/128+1/256
=> 2A = 1+1/2+1/4+1/8+1/18+1/32+1/64+1/128
=> 2A - A = 1 - 1/256
=> A = 255/256 nhé!
![](https://rs.olm.vn/images/avt/0.png?1311)
Ta có : \(\frac{1}{2}+\frac{1}{6}+\frac{1}{12}+\frac{1}{20}+\frac{1}{30}+\frac{1}{42}\)
\(=\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+\frac{1}{4.5}+\frac{1}{5.6}+\frac{1}{6.7}\)
\(=1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+......+\frac{1}{6}-\frac{1}{7}\)
\(=1-\frac{1}{7}=\frac{6}{7}\)
![](https://rs.olm.vn/images/avt/0.png?1311)
Ta có nhận xét sau
Với n là số nguyên dương thì
\(\frac{1}{n\left(n+1\right)}=\frac{1}{n}-\frac{1}{\left(n+1\right)}\)
Áp dụng vào , ta có
\(\frac{1}{2}+\frac{1}{6}+\frac{1}{12}=\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}=\frac{1}{1}-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}=1-\frac{1}{4}=\frac{3}{4}\)