Question

[Nguyễn Thị Thu Hải]

1/2+5/6+11/12+19/20+...+109/110

Answer 1

[Đặng Cường Thành]

Đặt biểu thức là A. A có 10 số hạng.

A = 1/2+5/6+11/12+19/20+...+109/110.

A = (1-1/2) + (1-1/6) + ...+(1-1/110)

A = 1+1+1+...+1(10 số 1) - (\(\frac{1}{2}\)+\(\frac{1}{6}\)+...+\(\frac{1}{110}\))

A=10-B

B = \(\frac{1}{1.2}\)+\(\frac{1}{2.3}\)+...+\(\frac{1}{10.11}\)

B = \(\frac{2-1}{1.2}\)+\(\frac{3-2}{2.3}\)+\(\frac{4-3}{3.4}\)+...+\(\frac{11-10}{10.11}\)

B=1-\(\frac{1}{2}\)+\(\frac{1}{2}\)-\(\frac{1}{3}\)+...+\(\frac{1}{10}\)-\(\frac{1}{11}\)

B=1-\(\frac{1}{11}\)=\(\frac{10}{11}\)

⇒A=10-B=10-\(\frac{10}{11}\)=\(\frac{100}{11}\)