\(C=\frac{1.2.3+2.3.4+3.4.5+4.5.6+5.6.7}{3.3.2.3.3.1+2.3.3.3.4.3+3.3.4.3.3.5+3.4.5.3.6.3+3.5.3.6.7.3}+\frac{8}{27}\)

\(C=\frac{1.2.3+2.3.4+3.4.5+4.5.6+5.6.7}{3^3.\left(1.2.3+2.3.4+3.4.5+4.5.6+5.6.7\right)}+\frac{8}{27}\)

\(C=\frac{1}{3^3}+\frac{8}{27}=\frac{1}{27}+\frac{8}{27}=\frac{9}{27}=\frac{1}{3}\)

Vậy C = \(\frac{1}{3}\)

Đặt A = 1.2.3 + 2.3.4 + 3.4.5 + ... + 28.29.30

4A = 1.2.3.(4-0) + 2.3.4.(5-1) + 3.4.5.(6-2) + ... + 28.29.30.(31-27)

4A = 1.2.3.4 - 0.1.2.3. + 2.3.4.5 - 1.2.3.4 + 3.4.5.6 - 2.3.4.5 + ... + 28.29.30.31 - 27.28.29.30

4A = 28.29.30.31 - 0.1.2.3

4A = 28.29.30.31

\(A=\frac{28.29.30.31}{4}=7.29.30.31=188790\)

Theo cách tính trên ta dễ dàng tính được:

1.2.3 + 2.3.4 + 3.4.5 + ... + (n - 1).n.(n + 1) = \(\frac{\left(n-1\right).n.\left(n+1\right).\left(n+2\right)}{4}\)

\(A=1\cdot2\cdot3+2\cdot3\cdot4+...+1998\cdot1999\cdot2000\)

\(\Rightarrow4A=1\cdot2\cdot3\cdot4+2\cdot3\cdot4\cdot4+....+1998\cdot1999\cdot2000\cdot4\)

\(\Rightarrow4A=1\cdot2\cdot3\cdot\left(4-0\right)+2\cdot3\cdot4\left(5-1\right)+....+1998\cdot1999\cdot2000\left(2001-1997\right)\)

\(\Rightarrow4A=1\cdot2\cdot3\cdot4-0\cdot1\cdot2\cdot3+2\cdot3\cdot4\cdot5-1\cdot2\cdot3\cdot4+...+1998\cdot1999\cdot2000\cdot2001-1997\cdot1998\cdot1999\cdot2000\)\(\Rightarrow4A=1997\cdot1998\cdot1999\cdot2000\)

\(\Rightarrow A=\frac{1997\cdot1998\cdot1999\cdot2000}{4}\)

A=1.2.3+2.3.4+3.4.5+...+112.113.114

4A= 1.2.3.4+ 3.4.5.4+....+112.113.114.4

4A= 1.2.3.(4-0)+3.4.5.(6-2)+.....+ 112.113.114.(115-111)

4A= 1.2.3.4 - 0.1.2.3 + 3.4.5.6 - 2.3.4.5 +.....+ 112.113.114.115 - 111.112.113.114

4A= 112.113.114.115=165920160

A=165920160:4=41480040

chắc chắn 100% đó

1.2.3+2.3.4+3.4.5+.......+112.113.114=1.2.3.4+2.3.4.4+........+112.113.114.4

                                                      =1.2.3.(5-1)+2.3.4.(6-1)+.................+112.113.114.(115-111)

                                                       =1.2.3-2.3.4+2.3.4-...........................-111.112.113+113.114.115

                                                       =113.114.115/3101

nho k minh nha

Đặt S = 1/1.2.3 - 1/2.3.4 - 1/3.4.5  - ...- 1/97.98.99

S x 2 = 2/1.2.3 - 2/2.3.4 - 2/3.4.5 - ...- 2/97.98.99

         = (1/1.2 -1/2.3) - (1/2.3 - 1/3.4 ) - (1/3.4 - 1/4.5) - ...- (1/97.98 - 1/98.99)

        = 1/1.2 - 1/2.3 - 1/2.3 + 1/3.4 - 1/3.4 + 1/4.5 - ....- 1/97.98 + 1/98.99

        = 1/2 -1/3 + 1/98.99

       =  1618/9072 => S = 1618/9072 : 2 = 809/9072

Câu f:

F = 1.2 + 2.3 + 3.4 + ... + n(n + 1)

F = 1[1.2.3 + 2.3.3 + 3.4.3 + ... + n.(n+1).3].1/3

F = [1.2.3 +2.3.(4-1) + 3.4.(5-2) +..+n(n+1).(n+2-n-1)].1/3

F = [1.2.3+2.3.4-1.2.3+...+n(n+1)(n+2)-(n-1).n.(n+1)].1/3

F= n.(n+1).(n+2)/3

g) G= 1.2.3+2.3.4+3.4.5+...+99.100.101

4G =1.2.3.4 + 2.3.4.4 + ...+99.100.101.4

4G =1.2.3.4 + 2.3.4.(5-1) + ...+99.100.101.(102-98)

4G = 1.2.3.4 +2.3.4.5- 1.2.3.4+...+99.100.101.102-98.99.100.101

4G = 99.100.101.102

G = 99.100.101.102/4

4(1.2.3) = 1.2.3.4 - 0.1.2.3

4(2.3.4) = 2.3.4.5 - 1.2.3.4

4(3.4.5) = 3.4.5.6 - 2.3.4.5

....................................

4(n-1)n(n+1) = (n-1)n(n+1)(n+2) - (n-2)(n-1)n(n+1)

=> 4 B = (n-1)n(n+1)(n+2) => B= (n-1)n(n+1)(n+2):4

4(1.2.3)=1.2.3.4 - 0.1.2.3

4(2.3.4)=2.3.4.5 - 1.2.3.4

4(3.4.5)=3.4.5.6 - 2.3.4.5

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