Question

\(Tínhtổng:\frac{1}{1x2x3}+\frac{1}{2x3x4}+...+\frac{1}{8x9x10}\)

Answer 1

\(\frac{1}{1.2.3}+\frac{1}{2.3.4}+...+\frac{1}{8.9.10}\)

\(=\frac{1}{1.2}-\frac{1}{2.3}+\frac{1}{2.3}-\frac{1}{3.4}+...+\frac{1}{8.9}-\frac{1}{9.10}\)

\(=\frac{1}{2}-\frac{1}{90}\)

\(=\frac{22}{45}\)

Answer 2

Gọi tổng trên là S , ta có :

S = 1/1.2.3 + 1/2.3.4 + ... + 1/8.9.10

S.2 = 2/1.2.3 + 1/2.3.4 + ... + 1/8.9.10

S.2 = 3 -1 /1.2.3 + 4 - 2/2.3.4 + ... + 10 - 8/8.9.10

S.2= 3/1.2.3 - 1/1.2.3 + 4/2.3.4 - 2/2.3.4 + ... + 10/8.9.10 - 8 /8.9.10

S.2 =1/1.2 - 1/2.3 + 1/2.3 - 1/3.4 + ... + 1/8.9 - 1/9.10

S.2 = 1/2 - 1/90

S = 1/4 - 1/360

S= 89/360

Answer 3

\(A=\frac{1}{1.2.3}+\frac{1}{2.3.4}+...+\frac{1}{8.9.10}\)

\(2A=\frac{2}{1.2.3}+\frac{2}{2.3.4}+...+\frac{2}{8.9.10}\)

\(2A=\frac{3-1}{1.2.3}+\frac{4-2}{2.3.4}+...+\frac{10-8}{8.9.10}\)

\(2A=\frac{3}{1.2.3}-\frac{1}{1.2.3}+\frac{4}{2.3.4}-\frac{2}{2.3.4}+...+\frac{10}{8.9.10}-\frac{8}{8.9.10}\)

\(2A=\frac{1}{1.2}-\frac{1}{2.3}+\frac{1}{2.3}-\frac{1}{2.4}+...+\frac{1}{8.9}-\frac{1}{9.10}\)

\(2A=\frac{1}{1.2}-\frac{1}{9.10}\)

\(2A=\frac{1}{2}-\frac{1}{90}=\frac{22}{45}\)

\(A=\frac{22}{45}:2=\frac{11}{45}\)

Answer 4

ta có:
4s=1.2.3.(4-0)+2.3.4.(5-1)+3.4.5.(6-2)+.........+k(k+1)(k+2)((k+3)-(k-1))
4s=1.2.3.4-1.2.3.0+2.3.4.5-1.2.3.4+3.4.5.6-2.3.4.5+........+k(k+1)(k+2)(k+3)-(k-1)k(k+1)(k+2)
4s=k(k+1)(k+2)(k+3)
ta biết rằng tích 4 số tự nhiên liên tiếp khi cộng thêm 1 luôn là 1 số chính phương
=>4s+1 là 1 số chính phương

Answer 5

\(\frac{1}{1\cdot2\cdot3}+\frac{1}{2\cdot3\cdot4}+...+\frac{1}{8\cdot9\cdot10}\)

\(=\left(\frac{2}{1\cdot2\cdot3}+\frac{2}{2\cdot3\cdot4}+...+\frac{2}{8\cdot9\cdot10}\right):2\)

\(=\left(\frac{1}{1\cdot2}-\frac{1}{2\cdot3}+\frac{1}{2\cdot3}-\frac{1}{3\cdot4}+...+\frac{1}{8\cdot9}-\frac{1}{9\cdot10}\right):2\)

\(=\left(\frac{1}{2}-\frac{1}{90}\right):2\)

\(=\frac{22}{45}\cdot\frac{1}{2}\)

\(=\frac{11}{45}\)

Answer 6

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