Đáp án:

a. 1/3

b. 3/7

c. 8/27

d. 31/15

e. 809/100

* Giải thích các bước giải:

a.

Đặt A = 1/2x3 + 1/3x4 + 1/4x5 + 1/5x6

⇒ A = 1/2 − 1/3 + 1/3 – 1/4 + 1/4 −1/5 + 1/5 – 1/6

⇒ A = 1/2 − 1/6

⇒ A = 1/3

b. 

Đặt B = 3/2x5 + 3/5.8 + 3/8.11 + 3/11.14

⇒ B = 1/2 – 1/5 + 1/5 – 1/8 + 1/8 – 1/11 + 1/11 – 1/14

⇒ B = 1/2 − 1/14

⇒ B = 7/14 – 1/14

⇒ B = 6/14 = 3/7

c. 

Đặt C = 4/3x7 - 4/7x11 + 4/11x15 - 4/15x19 + 4/19x23 - 4/23x27

⇒ C = 1/3 - 1/7 + 1/7 - 1/11 + 1/11 - 1/15 + 1/15 - 1/19 + 1/19 -

1/23 + 1/23 - 1/27 

⇒ C = 1/3 - 1/27

⇒ C =  9/27 - 1/27

⇒ C = 8/27

d. 2/3x5 + 2/5x7 + 2/7x9 - 2/9x11+ 2/11x13 + 2/13x15+ 2/1x2 +

2/2x3 + 2/3x4 +....+ 2/8x9 + 2/9x10

= (2/3x5 + 2/5x7 + 2/7x9 - 2/9x11+ 2/11x13 + 2/13x15) + (2/1x2

+ 2/2x3 + 2/3x4 +....+ 2/8x9 + 2/9x10)

= (1/3 – 1/5 + 1/5 − 1/7 + .... + 1/13 – 1/15) + [2 x (1 – 1/2 + 1/2 –

1/3 + .... +1/9 – 1/10)]

= (1/3 − 1/15) + [2 x (1−1/10)]

= (5/15 - 1/15) + [2 x (10/10 − 1/10)]

= (5/15 - 1/15) + (2 x 9/10)

= 4/15 + 9/5

= 31/15

e. 3/1x2 + 3/2x3 + 3/3x4 + 3/4x5 + 3/5x6 + ..... + 3/9x10 + 77/2x9

+ 77/9x16 + 77/ 16x23 + ... + 77/93x100

= (3/1x2 + 3/2x3 + 3/3x4 + 3/4x5 + 3/5x6 + ... + 3/9x10) +

(77/2x9 + 77/9x16 + 77/ 16x23 + ... + 77/93x100)

= [3/1 x (1/1 – 1/2 + 1/2 – 1/3 + 1//3 – 14 + .... + 1/9 – 1/10) ] +

[7/77 x (1/2 – 1/9 + 1/9 – 1/16 + 1/16  −1/23 + .... + 1/93 – 1/100)]

= [31 x (1/1 − 1/10)] + [7/77 x (1/2 − 1/100)]

= (3/1 x 9/10) + (7/77 x 49/100)

= 27/10 + 539/100

= 270/100 + 539/100

= 809/100