\(=\dfrac{1}{3}\left(\dfrac{1}{3}-\dfrac{1}{6}+\dfrac{1}{6}-\dfrac{1}{9}+...+\dfrac{1}{15}-\dfrac{1}{18}\right)\)

\(=\dfrac{1}{3}\left(\dfrac{1}{3}-\dfrac{1}{18}\right)=\dfrac{1}{3}\cdot\dfrac{5}{18}=\dfrac{5}{54}\)

\(\dfrac{5}{54}\)

3/5 A = 3/3.6 + 3/6.9 +..... + 3/96.99 

         = 1/3 - 1/6 + 1/6 - 1/9 + .... + 1/96 - 1/99 = 1/3 - 1/99 = 32/99

=> A = 160/297

k mk nha

Day ma la toan lop 5 a

đúng rồi đó

rồi,kp nha

\(\frac{4}{3.6}+\frac{4}{6.9}+...+\frac{4}{12.15}\)

\(=\frac{4\left(\frac{3}{3.6}+\frac{3}{6.9}+...+\frac{3}{12.15}\right)}{3}\)

\(=\frac{4\left(\frac{1}{3}-\frac{1}{6}+\frac{1}{6}-\frac{1}{9}+...+\frac{1}{12}-\frac{1}{15}\right)}{3}\)

\(=\frac{4\left(\frac{1}{3}-\frac{1}{15}\right)}{3}\)

\(=\frac{\frac{16}{15}}{3}=\frac{48}{15}\)

a/

\(1995^n.1997^n=\left(1995.1997\right)^n\)

\(1996^{2n}=\left(1996^2\right)^n\)

\(1995.1997=\left(1996-1\right).\left(1996+1\right)=1996^2-1\)

\(\Rightarrow1995.1997< 1996^2\Rightarrow1995^n.1997^n< 1996^{2n}\)

b/

\(A=\frac{1}{2.9}+\frac{1}{6.9}+\frac{1}{9.12}+\frac{1}{9.20}+\frac{1}{9.30}+\frac{1}{9.42}+\frac{1}{9.56}\)

\(A=\frac{1}{9}\left(\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+\frac{1}{4.5}+\frac{1}{5.6}+\frac{1}{6.7}+\frac{1}{7.8}\right)\)

\(A=\frac{1}{9}\left(\frac{2-1}{1.2}+\frac{3-2}{2.3}+\frac{4-3}{3.4}+...+\frac{8-7}{7.8}\right)\)

\(A=\frac{1}{9}\left(1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{7}-\frac{1}{8}\right)\)

\(A=\frac{1}{9}\left(1-\frac{1}{9}\right)=\frac{1}{9}.\frac{8}{9}=\frac{8}{81}\)

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

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

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

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

\(A=\frac{100}{101}\div2=\frac{50}{101}\)

b) Đặt \(B=\frac{1}{3.6}+\frac{1}{6.9}+\frac{1}{9.12}+\frac{1}{12.15}\)

\(3B=\frac{3}{3.6}+\frac{3}{6.9}+\frac{3}{9.12}+\frac{3}{12.15}\)

\(3B=\frac{1}{3}-\frac{1}{6}+\frac{1}{6}-\frac{1}{9}+\frac{1}{9}-\frac{1}{12}+\frac{1}{12}-\frac{1}{15}\)

\(3B=\frac{1}{3}-\frac{1}{15}=\frac{4}{15}\)

\(B=\frac{4}{15}\div3=\frac{4}{45}\)

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

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

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

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

\(A=\frac{100}{101}\div2=\frac{50}{101}\)

\(N=\dfrac{2}{3.6}+\dfrac{2}{6.9}+...+\dfrac{2}{2019.2022}\)

\(\Rightarrow N=2\left(\dfrac{1}{3.6}+\dfrac{1}{6.9}+...+\dfrac{1}{2019.2022}\right)\)

\(\Rightarrow N=\dfrac{2}{3}\left(\dfrac{3}{3.6}+\dfrac{3}{6.9}+...+\dfrac{3}{2019.2022}\right)\)

\(\Rightarrow N=\dfrac{2}{3}\left(\dfrac{1}{3}-\dfrac{1}{6}+\dfrac{1}{6}-\dfrac{1}{9}+...+\dfrac{1}{2019}-\dfrac{1}{2022}\right)\)

\(\Rightarrow N=\dfrac{2}{3}\left(\dfrac{1}{3}-\dfrac{1}{2022}\right)\)

\(\Rightarrow N=\dfrac{2}{3}.\dfrac{673}{2022}\\ \Rightarrow N=\dfrac{673}{3033}\)

Đặt \(A=\frac{3}{3\cdot6}+\frac{3}{6\cdot9}+\frac{3}{9\cdot12}+...+\frac{3}{96\cdot99}\)

\(\Rightarrow A=\frac{1}{3}-\frac{1}{6}+\frac{1}{6}-\frac{1}{9}+\frac{1}{9}-\frac{1}{12}+...+\frac{1}{96}-\frac{1}{99}\)

\(\Rightarrow A=\frac{1}{3}-\frac{1}{99}\)

\(\Rightarrow A=\frac{32}{99}\)

\(\frac{3}{3.6}+\frac{3}{6.9}+\frac{3}{9.12}+...+\frac{3}{96.99}\)

\(=\frac{3}{3}.\left(\frac{1}{3}-\frac{1}{6}+\frac{1}{6}-\frac{1}{9}+\frac{1}{9}-\frac{1}{12}+...+\frac{1}{96}-\frac{1}{99}\right)\)

\(=1.\left(\frac{1}{3}-\frac{1}{99}\right)\)

\(=1.\frac{32}{99}\)

\(=\frac{32}{99}\)