A.3=5.6.3+6.7.3+...+30.31.3

           A.3=5.6.(7-4)+6.7.(8-5)+...+30.31.(32-29)

          A.3=5.6.7-4.5.6+6.7.8-5.6.7+...+30.31.32-29.30.31

          A.3=(5.6.7-5.6.7)+...+(29.30.31-29.30.31)+(30.31.32-4.5.6)

         A.3=0+...+0+30.31.32-4.5.6

         A.3=30.31.32-4.5.6

        A=30.31.32-4.5.6 /3

        A=(29760-120)/3

       A=29460/3

        A=9880

  vậy A là 9880

 lưa ý dấu này/ nghĩa là chia

3.A = 5.6.(7-4) + 6.7.(8-5) + ....+30.31.(32- 29)

3.A = 5.6.7 - 4.5.6 + 6.7.8 - 5.6.7 + ...+ 30.31.32 - 29.30.31

3.A = (5.6.7 + 6.7.8 + ...+ 30.31.32) - (4.5.6 + 5.6.7 + ...+ 29.30.31)

3.A = 30.31.32 - 4.5.6 = 29 640 => A = 9 880  

Câu 1: 

Đặt S = 1.2+2.3+3.4+...+30.31

  3 S  = 1.2.3+2.3.3+3.4.3+...+30.31.3

  3 S  = 1.2.(3-0) + 2.3.(4-1) + 3.4.(5-2) + ...+ 30.31.(32-29)

  3S  = 1.2.3 + 2.3.4-2.3 + 3.4.5-2.3.4 + ...+ 30.31.32-29.30.31

  3S= 30.31.32 

S= 30.31.32/3

Bài 1:

$A=1.2+2.3+3.4+...+201.202$

$3A=1.2.3+2.3(4-1)+3.4(5-2)+....+201.202(203-200)$

$=1.2.3+2.3.4-1.2.3+3.4.5-2.3.4+....+201.202.203-200.201.202$

$=(1.2.3+2.3.4+3.4.5+...+201.202.203)-(1.2.3+2.3.4+....+200.201.202)$

$=201.202.203$

$\Rightarrow A=\frac{201.202.203}{3}=2747402$

Bài 2:

$S=4.5+5.6+6.7+....+100.101$

$3S=4.5(6-3)+5.6.(7-4)+6.7.(8-5)+....+100.101(102-99)$

$=4.5.6-3.4.5+5.6.7-4.5.6+6.7.8-5.6.7+....+100.101.102-99.100.101$

$=(4.5.6+5.6.7+6.7.8+...+100.101.102)-(3.4.5+4.5.6+5.6.7+...+99.100.101)$

$=100.101.102-3.4.5$

$\Rightarrow S=\frac{100.101.102-3.4.5}{3}=343380$

Bài 5:

a) Ta có: \(A=1\cdot2+2\cdot3+3\cdot4+...+9\cdot10\)

\(\Leftrightarrow3\cdot A=3\cdot\left(1\cdot2+2\cdot3+3\cdot4+...+9\cdot10\right)\)

\(\Leftrightarrow3A=1\cdot2\cdot\left(3-0\right)+2\cdot3\cdot\left(4-1\right)+3\cdot4\cdot\left(5-2\right)+...+9\cdot10\cdot\left(11-8\right)\)

\(\Leftrightarrow3A=1\cdot2\cdot3+2\cdot3\cdot4-1\cdot2\cdot3+3\cdot4\cdot5-2\cdot3\cdot4+...+8\cdot9\cdot10-8\cdot9\cdot10+9\cdot10\cdot11\)

\(\Leftrightarrow3\cdot A=9\cdot10\cdot11=90\cdot11=990\)

hay A=330

Vậy: A=330