tao có:

2p=2/1.2.3+2/2.3.4+...+2/n.n(+1)n(n+2)

2p=3-1/1.2.3+4-2/1.2.3+...+(n+2)-n/n.(n+1).(n+2)

2p=3/1.2.3-1/1.2.3+4/2.3.4-2/2.3.4+...+(n+2)/n.(n+1).(n+2)-n/n.(n+1).(n+2)

2p=1/1.2-1/2.3+1/2.3-1/3.4+...+1/n.(n+1)-1/(n+1).(n+2)

2p=1/1.2-1/(n+1).(n+2)

2p=(n+!).(n+2)-2/(2n+2).(n+2)

suy ra p=(n+1).(n+2)-2/(2n+2).(2n+4)

2s=3-1/1.2.3+4-2/1.2.3+...+50-48/48.49.50

2s=3/1.2.3-1/1.2.3+4/2.3.4-2/2.3.4+...+50/49.50.48-48/48.50.49

2s=1/1.2-1/2.3+1/2.3-1/3.4+...+1/48.49-1/49.50

2s=1/1.2-1/49.50

'2s=1/2-1/2450

2s=1225/2450-1/2450

2s=1224/2450

s=612/1225

\(P=\frac{1}{1\cdot2\cdot3}+\frac{1}{2\cdot3\cdot4}+\frac{1}{3\cdot4\cdot5}+...+\frac{1}{n\left(n+1\right)\left(n+2\right)}\)1

\(P=\frac{1}{2}\left(\frac{2}{1\cdot2\cdot3}+\frac{2}{2\cdot3\cdot4}+\frac{2}{3\cdot4\cdot5}+...+\frac{2}{n\left(n+1\right)\left(n+2\right)}\right)\)

\(P=\frac{1}{2}\left(\frac{1}{1\cdot2}-\frac{1}{2\cdot3}+\frac{1}{2\cdot3}-\frac{1}{3\cdot4}+\frac{1}{3\cdot4}-\frac{1}{4\cdot5}+...+\frac{1}{n\left(n+1\right)}-\frac{1}{\left(n+1\right)\left(n+2\right)}\right)\)

\(P=\frac{1}{2}\left(\frac{1}{2}-\frac{1}{\left(n+1\right)\left(n+2\right)}\right)\)

\(P=\frac{\left(\frac{1}{2}-\frac{1}{\left(n+1\right)\left(n+2\right)}\right)}{2}\)

S cx tinh giong v

\(S=\dfrac{1}{1.2.3}+\dfrac{1}{2.3.4}+...+\dfrac{1}{78.79.80}\)

\(\Rightarrow S=\dfrac{79\left(79+3\right)}{4\left(79+1\right)\left(79+2\right)}\)

\(S=\dfrac{79.82}{4.80.81}=\dfrac{79.41}{160.81}=\dfrac{3239}{12960}\)

\(S=\frac{1}{1.2.3}+\frac{1}{2.3.4}+...+\frac{1}{18.19.20}\)

\(2S=\frac{2}{1.2.3}+\frac{2}{2.3.4}+...+\frac{2}{18.19.20}\)

\(=\left(\frac{1}{1.2}-\frac{1}{2.3}\right)+\left(\frac{1}{2.3}-\frac{1}{3.4}\right)+...+\left(\frac{1}{18.19}+\frac{1}{19.20}\right)\)

\(=\frac{1}{2}-\frac{1}{2.3}+\frac{1}{2.3}-\frac{1}{3.4}+...+\frac{1}{18.19}-\frac{1}{19.20}\)

\(=\frac{1}{2}-\frac{1}{19.20}< \frac{1}{2}\)

\(2S< \frac{1}{2}\)

\(\Rightarrow S< \frac{1}{4}\) (ĐPCM)

\(S=\frac{1}{1.2.3}+\frac{1}{2.3.4}+...+\frac{1}{18.19.20}\)

    \(=\frac{1}{2}\left(\frac{2}{1.2.3}+\frac{2}{2.3.4}+...+\frac{2}{18.19.20}\right)\)

     \(=\frac{1}{2}\left(\frac{1}{1.2}-\frac{1}{2.3}+\frac{1}{2.3}-\frac{1}{3.4}+...+\frac{1}{18.19}-\frac{1}{19.20}\right)\)

       \(=\frac{1}{2}\left(\frac{1}{2}-\frac{1}{19.20}\right)\)

         \(=\frac{1}{2}\left(\frac{1}{2}-\frac{1}{380}\right)\)

           \(=\frac{1}{4}-\frac{1}{760}\)

=> S < \(\frac{1}{4}\)( vì 1/4 > 0)

Ta có:\(\frac{1}{n\left(n+1\right)\left(n+2\right)}=\frac{2+n-n}{2n\left(n+1\right)\left(n+2\right)}\)

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

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

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

Áp dụng:

\(S=\frac{1}{2}\left(\frac{1}{1.2}-\frac{1}{2.3}+\frac{1}{2.3}-\frac{1}{3.4}+...+\frac{1}{18.19}-\frac{1}{19.20}\right)\)

\(=\frac{1}{2}\left(\frac{1}{1.2}-\frac{1}{19.20}\right)=\frac{1}{4}-\frac{1}{2.19.20}< \frac{1}{4}\left(dpcm\right)\)

S = 1/1.2.3 + 1/2.3.4 + 1/3.4.5 + ...  + 1/23.24.25

2.S = 2/1.2.3 + 2/2.3.4 + 2/3.4.5 + ...  + 2/23.24.25

      = 1/1.2 - 1/2.3 + 1/2.3 - 1/3.4 + ...+ 1/23.24 - 1/24.25

      = 1/1.2 - 1/24.25 = 1/2 - 1/600

=> S = (1/2 - 1/600) : 2 = 1/4 - 1/1200

Dễ thấy S < 1/4 Hay S < 0,25

1/2.(2/1.2.3+2/2.3.4+......2/23.24.25)

1/2.(1/1.2-1/2.3+1/2.3-1/3.4+……+1/23.24-1/24.25)

1/2.(1/1.2-1/24.25)

1/2.(1/2-1/600)

1/2.(300/600-1/600)

1/2.299/600

299/1200

Ta co 0.25=1/4

Nen ta so sanh 1/4 va 299/1200

Vi 300/1200>299/1200

Nen 1/4>299/1200

Ket luan 0,25>S

S= 

1.2.3

1

​

 + 

2.3.4

1

​

 +...+ 

(n−1).n.(n+1)

1

​

 +...+ 

23.24.25

1

​

=

1

2

.

(

1

1.2

−

1

2.3

+

1

2.3

−

1

3.4

+

.

.

.

+

1

(

�

−

1

)

.

�

−

1

�

.

(

�

+

1

)

+

.

.

.

+

1

23.24

−

1

24.25

)

= 

2

1

​

 .( 

1.2

1

​

 − 

2.3

1

​

 + 

2.3

1

​

 − 

3.4

1

​

 +...+ 

(n−1).n

1

​

 − 

n.(n+1)

1

​

 +...+ 

23.24

1

​

 − 

2-gameguardian .25

1

​

 )

=

1

2

.

(

1

1.2

−

1

24.25

)

=

299

1200

= 

2

1

​

 .( 

1.2

1

​

 − 

24.25

1

​

 )= 

1200

299

b) S = \(\frac{1}{1.2.3}+\frac{1}{2.3.4}+...+\frac{1}{98.99.100}\)

\(=\frac{1}{2}.\left(\frac{2}{1.2.3}+\frac{2}{2.3.4}+...+\frac{2}{98.99.100}\right)\)

\(=\frac{1}{2}.\left(\frac{1}{1.2}-\frac{1}{2.3}+\frac{1}{2.3}-\frac{1}{3.4}+...+\frac{1}{98.99}-\frac{1}{99.100}\right)\)

\(=\frac{1}{2}.\left(\frac{1}{1.2}-\frac{1}{99.100}\right)\)

\(=\frac{1}{2}.\left(\frac{1}{2}-\frac{1}{9900}\right)\)

\(=\frac{1}{2}.\frac{4949}{9900}\)

\(=\frac{4949}{19800}\)

\(2S=\dfrac{2}{1.2.3}+\dfrac{2}{2.3.4}+...+\dfrac{2}{23+24+25}=\left(\dfrac{1}{1.2}-\dfrac{1}{2.3}\right)+\left(\dfrac{1}{2.3}-\dfrac{1}{3.4}\right)+...+\left(\dfrac{1}{23.24}-\dfrac{1}{24.25}\right)\)\(=\dfrac{1}{1.2}-\dfrac{1}{24.25}=\dfrac{299}{600}\) 

Vậy \(S=\dfrac{299}{600}\div2=\dfrac{299}{1200}\)

bài này dễ quá

tui biết = mấy 

kb và mỗi ngày k 3 nick nguyên huyên 

                                       sakura

                                       Nguyễn Thị Thu Huyền 

thì tui giải cho

S_n= 1/1.2.3+1/2.3.4+1/3.4.5+...+1/n(n+1)(n+2)

2S_n= 2/1.2.3+2/2.3.4+2/3.4.5+...+2/n(n+1)(n+2)

2S_n= 1/1.2-1/2.3+1/2.3-1/3.4+1/3.4-1/4.5+...+1/n(n+1)-1/(n+1)(n+2)

2S_n= 1/1.2-1/(n+1)(n+2)

2S_n= (n+1)(n+2)-2/2(n+1)(n+2)

S_n= (n+1)(n+2)-2/4(n+1)(n+2)

\(A=\dfrac{1}{1.2.3}+\dfrac{1}{2.3.4}+....+\dfrac{1}{2014.2015.2016}\\ =\dfrac{1}{2}\left(\dfrac{1}{1.2}-\dfrac{1}{2.3}\right)+\left(\dfrac{1}{2.3}-\dfrac{1}{3.4}\right)+.....+\left(\dfrac{1}{2014.2015}-\dfrac{1}{2015.2016}\right)\\ =\dfrac{1}{2}\left(\dfrac{1}{1.2}-\dfrac{1}{2015.2016}\right)\\ =\dfrac{1}{4}-\dfrac{1}{2.2015.2016}< \dfrac{1}{4}\\ \Rightarrow A< \dfrac{1}{4}\)

Giải:

Ta có: \(A=\dfrac{1}{1.2.3}+\dfrac{1}{2.3.4}+...+\dfrac{1}{2014.2015.2016}\)

\(=\dfrac{1}{2}\left(\dfrac{2}{1.2.3}+\dfrac{2}{2.3.4}+...+\dfrac{2}{2014.2015.2016}\right)\)

\(=\dfrac{1}{2}\)\(\left(\dfrac{1}{1.2}-\dfrac{1}{2.3}+...+\dfrac{1}{2014.2015}-\dfrac{1}{2015.2016}\right)\)

\(=\dfrac{1}{2}\left(\dfrac{1}{1.2}-\dfrac{1}{2015.2016}\right)\)

\(=\dfrac{1}{2}.\dfrac{1}{1.2}-\dfrac{1}{2}.\dfrac{1}{2015.2016}=\dfrac{1}{4}-\) \(\dfrac{1}{2.2015.2016}\)

Mà \(\dfrac{1}{2.2015.2016}>0\Leftrightarrow\dfrac{1}{4}-\dfrac{1}{2.2015.2016}< \dfrac{1}{4}\)

Vậy \(A< \dfrac{1}{4}\)

\(A=\dfrac{1}{1\cdot2\cdot3}+\dfrac{1}{2\cdot3\cdot4}+...+\dfrac{1}{2014\cdot2015\cdot2016}\)

\(=\dfrac{1}{2}\left(\dfrac{2}{1\cdot2\cdot3}+\dfrac{2}{2\cdot3\cdot4}+...+\dfrac{2}{2014\cdot2015\cdot2016}\right)\)

\(=\dfrac{1}{2}\left(\dfrac{1}{1\cdot2}-\dfrac{1}{2\cdot3}+\dfrac{1}{2\cdot3}-\dfrac{1}{3\cdot4}+...+\dfrac{1}{2014\cdot2015}-\dfrac{1}{2015\cdot2016}\right)\)

\(=\dfrac{1}{2}\left(\dfrac{1}{1\cdot2}-\dfrac{1}{2015\cdot2016}\right)\)

\(=\dfrac{1}{4}-\dfrac{1}{8124480}< \dfrac{1}{4}\)

\(\Rightarrow A< \dfrac{1}{4}\)

Vậy \(A< \dfrac{1}{4}\).

Ta có :

\(S=\dfrac{1}{1.2.3}+\dfrac{1}{2.3.4}+..............+\dfrac{1}{98.99.100}\)

\(S=\dfrac{1}{2}\left(\dfrac{2}{1.2.3}+\dfrac{2}{2.3.4}+................+\dfrac{2}{98.99.100}\right)\)

\(S=\dfrac{1}{2}\left(\dfrac{1}{1.2}-\dfrac{1}{2.3}+\dfrac{1}{2.3}-\dfrac{1}{3.4}+...........+\dfrac{1}{98.99}-\dfrac{1}{99.100}\right)\)

\(S=\dfrac{1}{2}\left(\dfrac{1}{1.2}-\dfrac{1}{99.100}\right)\)

\(S=\dfrac{1}{2}\left(\dfrac{1}{2}-\dfrac{1}{9900}\right)\)

\(S=\dfrac{1}{2}.\dfrac{4949}{9900}\)

\(S=\dfrac{4949}{19800}\)

~ Chúc bn học tốt ~