\(A=\frac{\frac{2000\cdot2001\cdot2002\cdot...\cdot2999}{1\cdot2\cdot3\cdot...\cdot1000}}{\frac{1001\cdot1002\cdot1003\cdot...\cdot2999}{1\cdot2\cdot3\cdot...\cdot1999}}=\frac{2000\cdot2001\cdot2002\cdot...\cdot2999}{1\cdot2\cdot3\cdot...\cdot1000}\times\frac{1\cdot2\cdot3\cdot...\cdot1999}{1001\cdot1002\cdot1003\cdot...\cdot2999}\)
\(A=1\)
Tính:
\(\left(\frac{1000}{1}+\frac{999}{2}+\frac{998}{3}+\frac{997}{4}+...+\frac{2}{999}+\frac{1}{1000}\right)\)\(:\)\(\left(\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+...+\frac{1}{1000}\right)\)
Giảu hệ phương trình (2000 ẩn số):
\(2x_1=x_2+\frac{1}{x_2}\left(1\right)\)
\(2x_2=x_3+\frac{1}{x_3}\left(2\right)\)
..................................
\(2x_{1999}=\frac{1}{x_{2000}}+x_{2000}\left(1999\right)\)
\(2x_{2000}=x_1+\frac{1}{x_1}\left(2000\right)\)
Chứng minh: \(\sqrt{\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{\left(a+b\right)^2}}=|\frac{1}{a}+\frac{1}{b}-\frac{1}{a+b}|\)
Áp dụng tính: M= \(\sqrt{1+999^2+\frac{999^2}{1000^2}}+\frac{999}{1000}\)
Tính : \(\frac{\left(1^4+\frac{1}{4}\right)\left(3^4+\frac{1}{4}\right)\left(5^4+\frac{1}{4}\right)\left(7^4+\frac{1}{4}\right)\left(9^4+\frac{1}{4}\right)\left(11^4+\frac{1}{4}\right)}{\left(2^4+\frac{1}{4}\right)\left(4^4+\frac{1}{4}\right)\left(6^4+\frac{1}{4}\right)\left(8^4+\frac{1}{4}\right)\left(10^4+\frac{1}{4}\right)\left(12^4+\frac{1}{4}\right)}\)
a) Chứng minh: \(\frac{1}{x}-\frac{1}{x+1}=\frac{1}{x\left(x+1\right)}\)
b). Tính nhẩm tổng sau: \(\frac{1}{x\left(x+1\right)}+\frac{1}{\left(x+1\right)\left(x+2\right)}+\frac{1}{\left(x+2\right)\left(x+3\right)}+\frac{1}{\left(x+3\right)\left(x+4\right)}+\frac{1}{\left(x+4\right)\left(x+5\right)}+\frac{1}{x+5}\)
TÍnh giá trị biểu thức
A=\(\frac{\left(1+\frac{1}{4}\right)\left(3^4+\frac{1}{4}\right)\left(5^4+\frac{1}{4}\right)...\left(29^4+\frac{1}{4}\right)}{\left(2^4+\frac{1}{4}\right)\left(4^4+\frac{1}{4}\right)\left(6^4+\frac{1}{4}\right)...\left(30^4+\frac{1}{4}\right)}\)
Tính giá trị biểu thức
A=\(\frac{\left(1+\frac{1}{4}\right)\left(3^4+\frac{1}{4}\right)\left(5^4+\frac{1}{4}\right)..........\left(29^4+\frac{1}{4}\right)}{\left(2^4+\frac{1}{4}\right)\left(4^4+\frac{1}{4}\right)\left(6^4+\frac{1}{4}\right).........\left(30^4+\frac{1}{4}\right)}\)
Tính
\(A=\left(\frac{1}{2^2}-1\right).\left(\frac{1}{3^2}-1\right).\left(\frac{1}{4^2}-1\right)...\left(\frac{1}{98^2}-1\right).\left(\frac{1}{99^2}-1\right)\)
Thu gọn
\(A=\frac{\left(1^4+\frac{1}{4}\right)\left(3^4+\frac{1}{4}\right)\left(5^4+\frac{1}{4}\right)...\left(2009^4+\frac{1}{4}\right)}{\left(2^4+\frac{1}{4}\right)\left(4^4+\frac{1}{4}\right)\left(6^4+\frac{1}{4}\right)...\left(2010^4+\frac{1}{4}\right)}\)
\(B=\frac{\left(a+2008\right)!+\left(a+2009\right)!}{\left(a+2008\right)!-\left(a+2009!\right)}\)
