Question

1) Tính

a)\(\dfrac{1}{1.3}+\dfrac{1}{3.5}+\dfrac{1}{5.7}+...+\dfrac{1}{99.101}\)

b)\(\dfrac{1}{4}+\dfrac{1}{32}+\dfrac{1}{96}+\dfrac{1}{192}\)(Điều kiện: không quy đồng và máy tính bỏ túi)

Answer 1

Đợi mình được không mình làm được câu a

Answer 2

a)\(\dfrac{1}{2}\).(\(\dfrac{2}{1.3}\)+\(\dfrac{2}{3.5}\)+\(\dfrac{2}{5.7}\)+........+\(\dfrac{2}{99.101}\))

=\(\dfrac{1}{2}\).(1-\(\dfrac{1}{3}\)+\(\dfrac{1}{3}\)-\(\dfrac{1}{5}\)+........+\(\dfrac{1}{99}\)-\(\dfrac{1}{101}\))

=\(\dfrac{1}{2}\).(1-\(\dfrac{1}{101}\))

=\(\dfrac{1}{2}\).\(\dfrac{100}{101}\)

=\(\dfrac{100}{202}\)=\(\dfrac{50}{101}\)

Answer 3

a) Đặt \(\dfrac{1}{1.3}+\dfrac{1}{3.5}+\dfrac{1}{5.7}+....+\dfrac{1}{99.101}\) = A

Ta có:

A = \(\dfrac{1}{1.3}+\dfrac{1}{3.5}+\dfrac{1}{5.7}+....+\dfrac{1}{99.101}\)

\(\Rightarrow\) 2A = 2\(\left(\dfrac{1}{1.3}+\dfrac{1}{3.5}+\dfrac{1}{5.7}+....+\dfrac{1}{99.101}\right)\)

\(\Rightarrow\) 2A = \(\dfrac{2}{1.3}+\dfrac{2}{3.5}+\dfrac{2}{5.7}+....+\dfrac{2}{99.101}\)

\(\Rightarrow\) 2A = \(1-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{5}+\dfrac{1}{5}-\dfrac{1}{7}+....+\dfrac{1}{99}-\dfrac{1}{101}\)

\(\Rightarrow\) 2A = 1 - \(\dfrac{1}{101}\) = \(\dfrac{100}{101}\)

\(\Rightarrow\) A = \(\dfrac{100}{101}\) : 2

Answer 4

\(a,\dfrac{1}{1.3}+\dfrac{1}{3.5}+\dfrac{1}{5.7}+...+\dfrac{1}{99.101}=\dfrac{1}{2}.\left(\dfrac{2}{1.3}+\dfrac{2}{3.5}+\dfrac{2}{5.7}+...+\dfrac{2}{99.101}\right)=\dfrac{1}{2}.\left(1-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{5}+\dfrac{1}{5}-\dfrac{1}{7}+...+\dfrac{1}{99}-\dfrac{1}{101}\right)=\dfrac{1}{2}.\left(1-\dfrac{1}{101}\right)=\dfrac{1}{2}.\left(\dfrac{101}{101}-\dfrac{1}{101}\right)=\dfrac{1}{2}.\dfrac{100}{101}=\dfrac{1.100}{2.101}=\dfrac{1.50}{1.101}=\dfrac{50}{101}\)\(b,\dfrac{1}{4}+\dfrac{1}{32}+\dfrac{1}{96}+\dfrac{1}{192}=\dfrac{1}{1.4}+\dfrac{1}{4.8}+\dfrac{1}{8.12}+\dfrac{1}{12.16}=\dfrac{1}{3}.\left(\dfrac{3}{1.4}+\dfrac{3}{4.8}+\dfrac{3}{8.12}+\dfrac{3}{12.16}\right)=\dfrac{1}{3}.\left(1-\dfrac{1}{4}+\dfrac{1}{4}-\dfrac{1}{8}+\dfrac{1}{8}-\dfrac{1}{12}+\dfrac{1}{12}-\dfrac{1}{16}\right)=\dfrac{1}{3}.\left(1-\dfrac{1}{16}\right)=\dfrac{1}{3}.\dfrac{15}{16}=\dfrac{1.15}{3.16}=\dfrac{1.5}{1.16}=\dfrac{5}{16}\)Chúc bạn học tốt, thành công trong học tập.