Đặt A = 1.2.3 + 2.3.4 + 3.4.5 +...+ (n - 1)n(n + 1)

4A = 1.2.3.4 + 2.3.4.4 + 3.4.5.4 +...+ (n - 1)n(n + 1).4

4A = 1.2.3.(4 - 0) + 2.3.4.(5 - 1) + 3.4.5.(6 - 2) +....+ (n - 1)n(n + 1).[(n + 2) - (n - 2)]

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 +...+ (n - 1)n(n + 1)(n + 2) - (n - 2)(n - 1)n(n + 1)

4A = [1.2.3.4 + 2.3.4.5 + 3.4.5.6 +....+ (n - 1)n(n + 1)(n + 2)] - [0.1.2.3 + 1.2.3.4 + 2.3.4.5 + (n - 2)(n - 1)n(n + 1)]

4A = (n - 1)n(n + 1)(n + 2) - 0.1.2.3

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

=> A = \(\frac{\left(n-1\right)n\left(n+1\right)\left(n+2\right)}{4}\)

=1/2-1/3-1/4+1/3-1/4-1/5+1/5-1/6-1/7+...+1/35-1/36-1/37

giao hoán, kết hợp là ra nha

Đặt \(A=\dfrac{1}{1\cdot2\cdot3}+\dfrac{1}{2\cdot3\cdot4}+\dfrac{1}{3\cdot4\cdot5}+...+\dfrac{1}{98\cdot99\cdot100}\)

Ta có: \(A=\dfrac{1}{1\cdot2\cdot3}+\dfrac{1}{2\cdot3\cdot4}+\dfrac{1}{3\cdot4\cdot5}+...+\dfrac{1}{98\cdot99\cdot100}\)

\(\Leftrightarrow2A=\dfrac{2}{1\cdot2\cdot3}+\dfrac{2}{2\cdot3\cdot4}+\dfrac{2}{3\cdot4\cdot5}+...+\dfrac{2}{98\cdot99\cdot100}\)

\(\Leftrightarrow2A=-\dfrac{1}{1\cdot2}+\dfrac{1}{2\cdot3}-\dfrac{1}{2\cdot3}+\dfrac{1}{3\cdot4}-\dfrac{1}{3\cdot4}+\dfrac{1}{4\cdot5}-\dfrac{1}{4\cdot5}+...-\dfrac{1}{98\cdot99}+\dfrac{1}{99\cdot100}\)

\(\Leftrightarrow2A=-\dfrac{1}{2}+\dfrac{1}{99\cdot100}\)

\(\Leftrightarrow2A=\dfrac{-1}{2}+\dfrac{1}{9900}\)

\(\Leftrightarrow2A=\dfrac{-4950}{9900}+\dfrac{1}{9900}=\dfrac{-4949}{9900}\)

hay \(A=\dfrac{-4949}{19800}\)

Câu b:

B = 1/1.2.3 + 1/2.3.4 + 1/3.4.5 + ...+ 1/98.99.100

B = 1/2. (2/1.2.3 + 2/2.3.4 + ...+ 2/98.99.100)

B = 1/2.(1/1.2 - 1/2.3 + 1/2.3 - 1/3.4 + ...+ 1/98.99 - 1/99.100)

B = 1/2.(1/2 - 1/9900)

B = 1/2.4949/9900

B = 4949/19800

+Câu a:

A = 1/1.2 + 1/2.3 + ...+ 1/5.6 + 1

A = 1/1 - 1/2 + 1/2 - 1/3 + ...+ 1/5 - 1/6 + 1

A = 1/1 - 1/6 + 1

A = 6/6 - 1/6 + 6/6

A = 5/6 + 6/6

A = 11/6

Câu b:

B = 1/1.2.3 + 1/2.3.4 + 1/3.4.5 + ...+ 1/98.99.100

B = 1/2. (2/1.2.3 + 2/2.3.4 + ...+ 2/98.99.100)

B = 1/2.(1/1.2 - 1/2.3 + 1/2.3 - 1/3.4 + ...+ 1/98.99 - 1/99.100)

B = 1/2.(1/2 - 1/9900)

B = 1/2.4949/9900

B = 4949/19800

Câu a:

1/1.2 + 1/2.3 + 1/3.4 + 1/4.5 + 1/5.6 + 1

= 1/1 - 1/2 + 1/2 - 1/3 + 1/4 - 1/5 + 1/5 - 1/6 + 1

= 1/1 - 1/6 + 1

= 6/6 - 1/6 + 6/6

= 5/6 + 1

= 11/6

Câu b:

Câu b:

B = 1/1.2.3 + 1/2.3.4 + 1/3.4.5 + ...+ 1/98.99.100

B = 1/2. (2/1.2.3 + 2/2.3.4 + ...+ 2/98.99.100)

B = 1/2.(1/1.2 - 1/2.3 + 1/2.3 - 1/3.4 + ...+ 1/98.99 - 1/99.100)

B = 1/2.(1/2 - 1/9900)

B = 1/2.4949/9900

B = 4949/19800