3M = 1.2.3 + 2.3.3 + 3.4.3 + 4.5.3 + ... + 212.213.3 
Mà : 
1.2.3 = 1.2.3 
2.3.3 = 2.3.4 - 2.3.1 
3.4.3 = 3.4.5 - 3.4.2 
4.5.3 = 4.5.6 - 4.5.3 
............................. 
............................. 
............................ 
211.212.3 = 211.212.213 - 211.212.210 
212.213.3 = 212.213.214 - 212.213.211 

Cộng tất cả lại vế theo vế ---> 3M = 212.213.214 
---> M = 212.213.214/3 = 212.71.214 = 3221128

sai đề

=> 3M = 1.2.3 + 2.3.3 + 3.4.3 + ... + 212.213.3

=> 3M = 1.2.( 3 - 0 ) + 2.3.( 4 - 1 ) + 3.4.( 5 - 2 ) + .... + 212.213.( 214 - 211 )

=> 3M = 1.2.3 + 2.3.4 - 1.2.3 + 3.4.5 - 2.3.4 + ... + 212.213.214 - 211.212.213

=> 3M = 212.213.214

=> M = ( 212.213.214 ) : 3

Đề hình như có chút sai sót?

Ko hiểu sao ở kia lại là 212 với 213 phải là a.b chứ

Gọi S = 1.2 + 2.3 + 3.4 + .... + 212.213 

=> 3S = 1.2.3 + 2.3.3 + 3.4.3 + ... + 212.213.3

=> 3S = 1.2.3 + 2.3.( 4 - 1 ) + 3.4.( 5 - 2 ) + ... + 212.213.( 214 - 211 )

=> 3S = 1.2.3 + 2.3.4 - 1.2.3 + 3.4.5 - 2.3.4 + .... + 212.213.214 - 211.212.213

=> 3S = 212.213.214

=> S = ( 212.213.214 ) : 3

Đặt A = 1.2 + 2.3 + 3.4 + ...... + 99.100

3A= 3.(1.2 + 2.3 + 3.4 + ..... +99.100)

3A=1.2.(3-0) + 2.3.(4-1) +.....+99.100.(101-98)

 3A=1.2.3 - 1.2.3 + 2.3.4 - 2.3.4  + .....+99.100.101

3A=99.100.101

A=99.100.101/3=333300

sưả đề \(P=1-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+...+\dfrac{1}{2021}-\dfrac{1}{2022}\)

\(=1-\dfrac{1}{2022}=\dfrac{2021}{2022}\)

\(A=4\left(1-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+...+\dfrac{1}{2014}-\dfrac{1}{2015}\right)\)

\(=4\cdot\dfrac{2014}{2015}=\dfrac{8056}{2015}\)

Trước tiên, chúng ta cần có lý thuyết về biến đổi phân số.

\(\dfrac{b-a}{a\cdot b}=\dfrac{1}{a}-\dfrac{1}{b}\)

Ta có:

\(S=\dfrac{1}{1\cdot2}+\dfrac{1}{2\cdot3}+\dfrac{1}{3\cdot4}+...+\dfrac{1}{2017\cdot2018}\)

\(S=1-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{4}+...+\dfrac{1}{2017}-\dfrac{1}{2018}\)

\(S=1+\left(-\dfrac{1}{2}+\dfrac{1}{2}\right)+\left(-\dfrac{1}{3}+\dfrac{1}{3}\right)+...-\dfrac{1}{2018}\)

\(S=1-\dfrac{1}{2018}\)

\(S=\dfrac{2017}{2018}\)

=1/1.2+1/2.3+1/3.4+...1/2017.2018

=1/1-1/2+1/2-1/3+1/3-1/4+...+1/2017-1/2018

=1-1/2018

=2018/2018-1/2018

=2017/2018

S = \(\dfrac{1}{1.2}\) + \(\dfrac{1}{2.3}\) + \(\dfrac{1}{3.4}\)+ .......+ \(\dfrac{1}{2017.2018}\)

S = \(\dfrac{1}{1}\) - \(\dfrac{1}{2}\) + \(\dfrac{1}{2}\) - \(\dfrac{1}{3}\) + \(\dfrac{1}{3}\) - \(\dfrac{1}{4}\)+.......+ \(\dfrac{1}{2017}\) - \(\dfrac{1}{2018}\)

S = \(\dfrac{1}{1}\) - \(\dfrac{1}{2018}\)

S = \(\dfrac{2017}{2018}\)