Bài 1:Tính giá trị biểu thức sau:
a,A=\(\dfrac{1}{2 . 3}\) + \(\dfrac{1}{3 . 4}\) + \(\dfrac{1}{4 . 5}\) + \(\dfrac{1}{5 . 6}\) + \(\dfrac{1}{6 . 7}\)
b,D=\(\dfrac{2}{3 . 4}\) + \(\dfrac{2}{4 . 5}\) + ... + \(\dfrac{2}{2022 . 2023}\)
c,E=\(\dfrac{1}{1 . 4}\) + \(\dfrac{1}{4 . 7}\) + ... + \(\dfrac{1}{37 . 40}\)
d,J=\(\dfrac{1 . 2 . 3 + 2 . 4 . 6 + 4 . 8 . 12 + 7 . 14 . 21}{1 . 3 . 5 + 2 . 6 . 10 + 4 . 12 . 20 + 7 . 21 . 35}\) + \(\dfrac{3}{5}\)
a) \(A=\dfrac{1}{2\cdot3}+\dfrac{1}{3\cdot4}+\dfrac{1}{4\cdot5}+\dfrac{1}{5\cdot6}+\dfrac{1}{6\cdot7}\)
\(A=\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{4}+\dfrac{1}{4}-\dfrac{1}{5}+\dfrac{1}{5}-\dfrac{1}{6}+\dfrac{1}{6}-\dfrac{1}{7}\)
\(A=\dfrac{1}{2}-\dfrac{1}{7}\)
\(A=\dfrac{5}{14}\)
b) \(D=\dfrac{2}{3\cdot4}+\dfrac{2}{4\cdot5}+...+\dfrac{2}{2022\cdot2023}\)
\(D=2\cdot\left(\dfrac{1}{3\cdot4}+\dfrac{1}{4\cdot5}+...+\dfrac{1}{2022\cdot2023}\right)\)
\(D=2\cdot\left(\dfrac{1}{3}-\dfrac{1}{4}+\dfrac{1}{4}-\dfrac{1}{5}+...+\dfrac{1}{2022}-\dfrac{1}{2023}\right)\)
\(D=2\cdot\left(\dfrac{1}{3}-\dfrac{1}{2023}\right)\)
\(D=2\cdot\dfrac{2020}{6069}\)
\(D=\dfrac{4040}{6069}\)
c) \(E=\dfrac{1}{1\cdot4}+\dfrac{1}{4\cdot7}+...+\dfrac{1}{37\cdot40}\)
\(E=\dfrac{1}{3}\cdot3\cdot\left(\dfrac{1}{1\cdot4}+\dfrac{1}{4\cdot7}+...+\dfrac{1}{37\cdot40}\right)\)
\(E=\dfrac{1}{3}\cdot\left(\dfrac{3}{1\cdot4}+\dfrac{3}{4\cdot7}+...+\dfrac{1}{37\cdot40}\right)\)
\(E=\dfrac{1}{3}\cdot\left(1-\dfrac{1}{4}+\dfrac{1}{4}-\dfrac{1}{7}+...+\dfrac{1}{37}-\dfrac{1}{40}\right)\)
\(E=\dfrac{1}{3}\cdot\left(1-\dfrac{1}{40}\right)\)
\(E=\dfrac{1}{3}\cdot\dfrac{39}{40}\)
\(E=\dfrac{13}{40}\)
d) \(J=\dfrac{1\cdot2\cdot3+2\cdot4\cdot6+4\cdot8\cdot12+7\cdot14\cdot21}{1\cdot3\cdot5+2\cdot6\cdot10+4\cdot12\cdot20+7\cdot21\cdot35}+\dfrac{3}{5}\)
\(J=\dfrac{\left(1\cdot2\cdot3\right)+2^3\cdot\left(1\cdot2\cdot3\right)+4^3\cdot\left(1\cdot2\cdot3\right)+7^3\cdot\left(1\cdot2\cdot3\right)}{\left(1\cdot3\cdot5\right)+2^3\cdot\left(1\cdot3\cdot5\right)+4^3\cdot\left(1\cdot3\cdot5\right)+7^3\cdot\left(1\cdot3\cdot5\right)}+\dfrac{3}{5}\)
\(J=\dfrac{1\cdot2\cdot3\cdot\left(1+2^3+4^3+7^3\right)}{1\cdot3\cdot5\cdot\left(1+2^3+4^3+7^3\right)}+\dfrac{3}{5}\)
\(J=\dfrac{1\cdot2\cdot3}{1\cdot3\cdot5}+\dfrac{3}{5}\)
\(J=\dfrac{2}{5}+\dfrac{3}{5}\)
\(J=1\)
\(a.\) \(A=\dfrac{1}{2\cdot3}+\dfrac{1}{3\cdot4}+\dfrac{1}{4\cdot5}+\dfrac{1}{5\cdot6}+\dfrac{1}{6\cdot7}\)
\(A=\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{4}+\dfrac{1}{4}-\dfrac{1}{5}+\dfrac{1}{5}-\dfrac{1}{6}+\dfrac{1}{6}-\dfrac{1}{7}\)
\(A=\dfrac{1}{2}-\dfrac{1}{7}\)
\(A=\dfrac{7}{14}-\dfrac{2}{14}\)
\(A=\dfrac{5}{14}\)
\(b.\) \(D=\dfrac{2}{3\cdot4}+\dfrac{2}{4\cdot5}+...+\dfrac{2}{2022\cdot2023}\)
\(D=\dfrac{2}{3}-\dfrac{2}{4}+\dfrac{2}{4}-\dfrac{2}{5}+...+\dfrac{2}{2022}-\dfrac{2}{2023}\)
\(D=\dfrac{2}{3}-\dfrac{2}{2023}\)
\(D=\dfrac{4046}{6069}-\dfrac{6}{6069}\)
\(D=\dfrac{4040}{6969}\)
\(c.\) \(E=\dfrac{1}{1\cdot4}+\dfrac{1}{4\cdot7}+...+\dfrac{1}{37\cdot40}\)
\(E=\dfrac{1}{1}-\dfrac{1}{4}+\dfrac{1}{4}-\dfrac{1}{7}+...+\dfrac{1}{37}-\dfrac{1}{40}\)
\(E=\dfrac{1}{4}-\dfrac{1}{40}\)
\(E=\dfrac{10}{40}-\dfrac{1}{40}\)
\(E=\dfrac{9}{40}\)
\(d.\) \(J=\dfrac{1\cdot2\cdot3+2\cdot4\cdot6+4\cdot8\cdot12+7\cdot14\cdot21}{1\cdot3\cdot5+2\cdot6\cdot10+4\cdot12\cdot20+7\cdot21\cdot35}+\dfrac{3}{5}\)
\(J=\dfrac{1\cdot2\cdot3+1\cdot2\cdot3\cdot2\cdot2\cdot2+1\cdot2\cdot3\cdot4\cdot4\cdot4+1\cdot2\cdot3\cdot7\cdot7\cdot7}{1\cdot3\cdot5+1\cdot2\cdot5\cdot2\cdot2\cdot2+1\cdot2\cdot5\cdot4\cdot4\cdot4+1\cdot2\cdot3\cdot7\cdot7\cdot7}+\dfrac{3}{5}\)
\(J=\dfrac{1\cdot2\cdot3\cdot\left(1+2^3+4^3+7^3\right)}{1\cdot3\cdot5\cdot\left(1+2^3+4^3+7^3\right)}+\dfrac{3}{5}\)
\(J=\dfrac{1\cdot2\cdot3}{1\cdot3\cdot5}\cdot1+\dfrac{3}{5}\)
\(J=\dfrac{6}{15}+\dfrac{3}{5}\)
\(J=\dfrac{2}{5}+\dfrac{3}{5}\)
\(J=\dfrac{5}{5}=1\)
