Question

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

Hãy so sánh S với 0,25 

Answer 1

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

Answer 2

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

Answer 3

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