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

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=

1

2

.

(

1

1.2

−

1

24.25

)

=

299

1200

= 

2

1

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 .( 

1.2

1

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 − 

24.25

1

​

 )= 

1200

299

A=1/1.2.3 + 1/2.3.4 + ...  + 1/23.24.25

2A=2/1.2.3 + 2/2.3.4 + ...  + 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

Tớ chỉ giải đến đó thôi còn lại các  bạn cứ bấm máy tính là ra

Bài toán trên áp dụng bài toán tổng quát sau:

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

Suy ra 

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

\(\frac{1}{1.2.3}+\frac{1}{2.3.4}+...+\frac{1}{23.24.25}\)

\(=\left(\frac{1}{1.2}-\frac{1}{2.3}+\frac{1}{2.3}-\frac{1}{3.4}+...+\frac{1}{23.24}-\frac{1}{24.25}\right).\frac{1}{2}\)

\(=\left(\frac{1}{1.2}-\frac{1}{24.25}\right).\frac{1}{2}\)

\(=\frac{299}{1200}\)

Giải :

 

= 1/2 . ( 3/1.2 - 3/2.3 + 3/2.3 - 3/3.4 + ... + 3/23.24 - 3/24.25 )

 

= 1/2 . ( 3/2 - 3/600 )

 

= 1/2 . 299/200

 

= 299/400

 

 Đúng(78)

\(S=\frac{1}{1.2.3}+\frac{1}{2.3.4}+...+\frac{1}{\left(n-1\right).n.\left(n+1\right)}+...+\frac{1}{23.24.25}\)

\(=\frac{1}{2}.\left(\frac{1}{1.2}-\frac{1}{2.3}+\frac{1}{2.3}-\frac{1}{3.4}+...+\frac{1}{\left(n-1\right).n}-\frac{1}{n.\left(n+1\right)}+...+\frac{1}{23.24}-\frac{1}{24.25}\right)\)

\(=\frac{1}{2}.\left(\frac{1}{1.2}-\frac{1}{24.25}\right)=\frac{299}{1200}\)

\(=\frac{1}{2}\left(\frac{2}{1.2.3}+\frac{2}{2.3.4}+...+\frac{2}{23.24.25}\right)=\frac{1}{2}\left(\frac{1}{1.2}-\frac{1}{2.3}+\frac{1}{2.3}-\frac{1}{3.4}+...+\frac{1}{23.24}-\frac{1}{24.25}\right)\)

\(=\frac{1}{2}\left(\frac{1}{2}-\frac{1}{600}\right)=\frac{1}{2}.\frac{299}{600}=\frac{299}{1200}\)

Giải :

= 1/2 . ( 3/1.2 - 3/2.3 + 3/2.3 - 3/3.4 + ... + 3/23.24 - 3/24.25 )

= 1/2 . ( 3/2 - 3/600 )

= 1/2 . 299/200

= 299/400

Giải: 

= 1/2 . ( 3/1.2 - 3/2.3 + 3/2.3 - 3/3.4 + .... + 3/23.24 - 3/24. 25 )

= 1/2 . ( 3/2 - 3/600)

=1/2 . 299/200

= 299/400

Giải:

= 1/2 . ( 3/1.2 - 3/2.3 + 3/2.3 - 3/3.4 + .... + 3/23.24 - 3/24. 25 )

= 1/2 . ( 3/2 - 3/600)

=1/2 . 299/200

= 299/400 

\(S=\frac{1}{1.2.3}+\frac{1}{2.3.4}+...+\frac{1}{23.24.25}\)

\(S=\frac{1}{2}\left(\frac{1}{1.2}-\frac{1}{2.3}+\frac{1}{2.3}-\frac{1}{3.4}+...+\frac{1}{23.24}-\frac{1}{24.25}\right)\)

\(S=\frac{1}{2}\left(\frac{1}{2}-\frac{1}{24.25}\right)\)

\(S=\frac{1}{4}-\frac{1}{24.50}\)

Dễ thấy với mọi số tự nhiên n > 1 , ta có :

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

Sử dụng  hệ thức trên cho từng số hạng trong tổng sau :

\(2S=\frac{2}{1.2.3}+\frac{2}{2.3.4}+...+\frac{2}{\left(n-1\right).n.\left(n+1\right)}+\frac{2}{23.24.25}\)

     \(=\frac{1}{1.2}-\frac{1}{2.3}+\frac{1}{2.3}-\frac{1}{3.4}+....+\frac{1}{\left(n-1\right).n}-\frac{1}{n.\left(n+1\right)}+...+\frac{1}{23.24}-\frac{1}{24.25}\)

Để ý rằng trong vế phải của hệ thức trên , trừ 2 số hạng đầu và cuối , các số hạng còn lại tạo thành từng cặp đối nhau.

Do đó , có thể rút gọn : 

\(2S=\frac{1}{1.2}-\frac{2}{24.25}=\frac{299}{600}\)

Vậy , ta được \(S=\frac{299}{600}\)

D = \(\frac{1}{1.2.3}+\frac{1}{2.3.4}+....+\frac{1}{37.38.39}\)

D = \(\frac{1}{2}.\left(\frac{3-1}{1.2.3}+\frac{4-2}{2.3.4}+...+\frac{39-37}{37.38.39}\right)\)

D = \(\frac{1}{2}.\left(\frac{3}{1.2.3}-\frac{1}{1.2.3}+\frac{4}{2.3.4}-\frac{2}{2.3.4}+...+\frac{39}{37.38.39}-\frac{37}{37.38.39}\right)\)

D = \(\frac{1}{2}.\left(\frac{1}{1.2}-\frac{1}{2.3}+\frac{1}{2.3}-\frac{1}{3.4}+....+\frac{1}{37.38}-\frac{1}{38.39}\right)\)

D = \(\frac{1}{2}.\left(\frac{1}{2}-\frac{1}{38.39}\right)\)

D = \(\frac{1}{2}.\frac{370}{741}\)

D = \(\frac{185}{741}\)

khó à nha