A = \(\dfrac{1}{1.2}+\dfrac{1}{2.3}+...+\dfrac{1}{49.50}\)

= \(\left(1-\dfrac{1}{2}\right)\)+\(\left(\dfrac{1}{2}-\dfrac{1}{3}\right)\)+...+\(\left(\dfrac{1}{49}-\dfrac{1}{50}\right)\)

= \(\left(1-\dfrac{1}{50}\right)\) = \(\dfrac{49}{50}\)

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

2E = \(\dfrac{2}{1.2.3}+\dfrac{2}{3.4.5}+...+\dfrac{2}{98.99.100}\)

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

= \(\dfrac{1}{1.2}\) - \(\dfrac{1}{99.100}\)= \(\dfrac{1}{2}-\dfrac{1}{9900}\) = \(\dfrac{4949}{9900}\)

b) \(\dfrac{1}{7}.\dfrac{5}{9}+\dfrac{5}{9}.\dfrac{2}{7}+\dfrac{5}{9}.\dfrac{1}{7}+\dfrac{5}{9}.\dfrac{3}{7}\)

= \(\dfrac{5}{9}.\left(\dfrac{1}{7}+\dfrac{2}{7}+\dfrac{1}{7}+\dfrac{3}{7}\right)\)

= \(\dfrac{5}{9}\) . 1= \(\dfrac{5}{9}\)

2) Tinh nhanh:

a) \(\dfrac{5}{23}\) . \(\dfrac{17}{26}\) + \(\dfrac{5}{23}\) . \(\dfrac{10}{26}\) - \(\dfrac{5}{23}\)

= \(\dfrac{5}{23}\) . \(\left(\dfrac{17}{26}+\dfrac{10}{26}-1\right)\)

= \(\dfrac{5}{23}\) . \(\left(\dfrac{27}{26}-1\right)\) = \(\dfrac{5}{23}\) . \(\dfrac{1}{26}\)

= \(\dfrac{5}{598}\)

a) Hai tia Ot và Oy cùng thuộc một nửa mặt phẳng có bờ chứa tia Ox, mà xÔt < xOy (40° <80°). Nên tia Ot nằm giữa hai tia Oy và Ox

b) Vì tia Ot nằm giữa hai tia Ox và Oy nên:

Ta có: xÔy = tÔx + tÔy

=> yÔt = xÔy — tÔx = 80° — 40° = 40°

Vậy tÔy = 40°

Vì tÔy = tÔx ( 40° = 40°)

Do đó tia Ot là tia phân giác của yÔx

— S = 1/4 + 2/4 +...+10/4 (1)

= 1 + 1/4 + 2/4 +...+ 9/4 (2)

=> Lấy (2) trừ đi (1) ta được:

1 — 10/4 = —6/4

Vì 14 = 14/1 = 84/6 mà —6/4 < 84/6

Do đó S < 14

c) C = 1/2 + 1/2^2 + 1/2^3 +...+ 1/2^10

=> 2C = 1 + 1/2 + 1/2^2 +...+ 1/2^9 (1)

= 1/2 + 1/2^2 +...+ 1/2^10

Lấy (1)-(2) ta được:

A = 1-1/2^10 = 1024/1024 - 1/1024 = 1023/1024

Chúc bạn học tốt

b) B = 1.2.3 + 2.3.4 + 3.4.5 +...+ 48.49.50

4B = 1.2.3.4 + 2.3.4.(5-1) + 3.4.5.(6-2) +...+ 48.49.50.(51-47)

4B-B = 1.2.3.4 + 2.3.4.(5-1) - 1.2.3.4 + 3.4.5.(6-2) - 2.3.4.(5-1) +...+ 48.49.50.(51-47) - 47.48.49.(50-46)

= 48.49.50.51:4 = 1499400

a) A = 1.3 +2.4 + 3.5 +...+ 97.99 + 98.100

A = 1(2 + 1) + 2(3+1) + 3(4 + 1) +...+ 98(99+1)

= (1.2 + 2.3 + 3.4 +...+ 98.99) + (1 + 2 + 3 +...+ 98)

= [ 1.2.3 + 2.3.(4-1) +...+ 98.99.(100-97)] + [ 1.2 + 2.(3-1) + 3.(4-2) +... 98.(99-97)]

= [ 1.2.3 + 2.3.(4-1) - 1.2.3 + 3.4.(5-2) - 2.3.(4-1) +...+ 98.99.(100-97) - 97.98(99-96)] + [ 1.2 + 2.(3-1) - 1.2 + 3.(4-2) - 2.(3-1) +...+ 98.(99-97) - 97(98-96)]

= 98.99.100:3 + 98.99:2 = 323 400 + 4581 = 328251