Ta có: \(A=\frac{1}{2\cdot4}+\frac{1}{4\cdot6}+...+\frac{1}{48\cdot50}\)

\(\Rightarrow2\cdot A=\frac{2}{2\cdot4}+\frac{2}{4\cdot6}+...+\frac{2}{48\cdot50}\)

\(\Rightarrow2\cdot A=\frac{1}{2}-\frac{1}{4}+\frac{1}{4}-\frac{1}{6}+...+\frac{1}{48}-\frac{1}{50}\)

\(\Rightarrow2\cdot A=\frac{1}{2}-\frac{1}{50}=\frac{12}{25}\)

\(\Rightarrow2\cdot A=\frac{12}{25}\)

hay \(A=\frac{12}{25}:2=\frac{12}{25}\cdot\frac{1}{2}=\frac{12}{50}=\frac{6}{25}\)

\(N=\dfrac{4}{2.4}+\dfrac{4}{4.6}+\dfrac{4}{6.8}+...+\dfrac{4}{2014.2016}\)

\(=2\left(\dfrac{2}{2.4}+\dfrac{2}{4.6}+\dfrac{2}{6.8}+...+\dfrac{2}{2014.2016}\right)\)

\(=2\left(\dfrac{1}{2}-\dfrac{1}{4}+\dfrac{1}{4}-\dfrac{1}{6}+\dfrac{1}{6}-\dfrac{1}{8}+...+\dfrac{1}{2014}-\dfrac{1}{2016}\right)\)

\(=2\left(\dfrac{1}{2}-\dfrac{1}{2016}\right)\)

\(=2\left(\dfrac{1008}{2016}-\dfrac{1}{2016}\right)\)

\(=2.\dfrac{1007}{2016}=\dfrac{1007}{1008}\)

Công thức đây bạn:

\(\dfrac{a}{n\left(n+a\right)}=\dfrac{1}{n}-\dfrac{1}{n+a}\)

Đặt \(A=2.4+4.6+...+50.52\)

\(\Rightarrow6A=2.4.6+4.6.6+...+50.52.6\)

\(=2.4.6+4.6.\left(8-2\right)+....+50.52.\left(54-48\right)\)

\(=2.4.5+4.6.8-2.4.6+...+50.52.54-48.50.52\)

\(=50.52.54\)

\(\Rightarrow A=\frac{50.52.54}{6}=23400\)

\(\text{Đặt A = 2.4 + 4.6 + ... + 50.52}\)

\(\text{⇒6A = 2.4.6 + 4.6.6 + ... + 50.52.6}\)

\(\text{6A= 2.4.6 + 4.6. 8 − 2 + .... + 50.52. 54 − 48 }\)

\(\text{6A= 2.4.5 + 4.6.8 − 2.4.6 + ... + 50.52.54 − 48.50.52}\)

\(\text{6A=50.52.54 }\)

\(\Rightarrow A=\)\(\frac{\text{50.52.54 }}{6}\)\(=23400\)

\(=\left(1+3+5+...+99+101\right)-\left(2+4+6+...98+100\right)\)
Thấy từ 1 đến 100 có (101-1)/2+1=51
=> 1+3+5+....+99+100=(1+101)x50/2=2601
Từ 2 đến 100 có (102-2)/2+1=50
=> 2+4+...+98+100=(2+100)X50/2=2550
=> D=2601-2550=51

2/2*4 + 2/4*6 + 3/6*8 + ... + 2/38*50

= 1/2 - 1/4 + 1/4 - 1/6 + 1/6 - 1/8 + .... + 1/38 - 1/50

= 1/2 - 1/50

= 24/50

=  12/25

\(\frac{1}{2.4}+\frac{1}{4.6}+\frac{1}{6.8}+...+\frac{1}{38.40}\)

=\(\frac{1}{2}-\frac{1}{4}+\frac{1}{4}-\frac{1}{6}+...+\frac{1}{38}-\frac{1}{40}\)

=\(\frac{1}{2}-\frac{1}{40}\)

=\(\frac{19}{40}\)

= 2 *[1/2 * 1/4 +1/4 *1/6 +1/6*1/8+...+1/38*1/40

=2*[1/2 - 1/40]

=2 * (-19/40)

= -380