\(A=1.\left(2+2\right)+2.\left(3+2\right)+3.\left(4+2\right)+....+99.\left(100+2\right)\)

\(A = (1.2 + 2.3 + 3.4 + ... + 99.100) + (1.2 + 2.2 + 3.2 + ... + 99.2)\)
\(Đặt B = 1.2 + 2.3 + 3.4 + ... + 99.100\)
\(3B = 1.2.(3-0) + 2.3.(4-1) + 3.4.(5-2) + ... + 99.100.(101-98)\)
\(3B = 1.2.3 - 0.1.2 + 2.3.4 - 1.2.3 + 3.4.5 - 2.3.4 + ... + 99.100.101 - 98.99.100\)
\(3B = 99.100.101\)
\(B = 33.100.101 = 333300\)
\(A = 333300 + 2.(1 + 2 + 3 + ... + 99)\)
\(A = 333300 + 2.(1 + 99).99:2\)
\(A = 333300 + 100.99\)
\(A = 333300 + 9900\)
\(A = 343200\)

a. A = 1.4 + 2.5 + 3.6 +...+ 99.102

       = 1( 2 +2) + 2(3+2) +...+ 99 (100 +2)

       = 1.2 + 1.2 +2.3 + 2.2 +...+ 99 .100 +99 . 2

       = ( 1.2 +2.3 + 3.4 +...+99 . 100) + 2(1 + 2 + 3+...+99)

       = 333300 + 9900 = 343 200

b. B = 1.3 + 2.4 + 3.5 +...+ 99.101

       = 1(2 +1) + 2(3 +1) + 3(4 +1) +...+ 99(100 +1)

       = 1.2 + 1 + 2.3 + 2 + 3.4 +...+ 99. 100 +99

       = ( 1.2 + 2.3 + 3.4 +...+ 99.100) + (1+2+...+99)

       = 333300 + 4950 = 338 250

c. C = 4 + 12 + 24 +...+ 19404 + 19800

  1/2C = 1.2 + 2.3 + 3.4 +...+ 98.99 + 99.100

1/2 C  = 333300 

    C   = 333300 : 1/2 = 666600

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

\(\Rightarrow3A=1.2.3+2.3\left(4-1\right)+3.4\left(5-2\right)+...+90.100\left(101-98\right)\)

\(\Rightarrow3A=1.2.3+2.3.4-1.2.3+3.4.5-2.3.4+...+99.100.101-98.99.100\)

\(\Rightarrow3A=99.100.101\)

\(\Rightarrow A=\left(99.100.101\right):3\)

\(\Rightarrow A=333300\)

\(B=1.3+2.4+3.5+...+99.101\)

\(\Rightarrow B=1\left(2+1\right)+2\left(3+1\right)+3\left(4+1\right)+...+99\left(100+1\right)\)

\(\Rightarrow B=1.2+1+2.3+2+3.4+3+...+99.100+99\)

\(\Rightarrow B=\left(1.2+2.3+3.4+...+99.100\right)+\left(1+2+3+...+99\right)\)

\(\Rightarrow B=333300+4950\)

\(\Rightarrow B=338250\)

\(D=1.2.3+2.3.4+3.4.5+...+98.99.100\)

\(\Rightarrow4D=1.2.3.4+2.3.4.4+...+98.99.100.4\)

\(\Rightarrow4D=1.2.3\left(4-0\right)+2.3.4\left(5-1\right)+...+98.99.100\left(101-97\right)\)

\(\Rightarrow4D=1.2.3.4-0.1.2.3+2.3.4.5-1.2.3.4+...+98.99.100-97.98.99\)

\(\Rightarrow4D=98.99.100\)

\(\Rightarrow D=\left(98.99.100\right):4=242550\)

A = 1(2+2)+2(3+2)+3(4+2)+.+99(100+2)

A = 1.2+1.2+2.3+2.2+3.4+3.2+.+99.100+99.2

A = (1.2+2.3+3.4+.+99.100)+2(1+2+3+.+99)

Phần còn lại bạn tự động não đi nha.

C= (4/1.3).(9/2.4).(16/3.5)...........(10000/99.101)
C=(4.9.16...........10000)/(1.3).(2.4)......(99.101)
C=(2^2.3^2.4^2..........100^2)/(1.2.3.4......99).(3.4.5.......101)
C=(2.3.4........100).(2.3.4......100)/(1.2.3.....99).(3.4.5....101)
Sau khi triệt tiêu ở tử và mẫu ta được:
C=(2.100)/101
C=200/101
 

tham the 

có giỏi thì làm một câu xem nào

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