Tính: \(\left(\dfrac{1000}{1}+\dfrac{999}{2}+\dfrac{998}{3}+...+\dfrac{2}{999}+\dfrac{1}{1000}\right):\left(\dfrac{1}{2}+\dfrac{1}{3}+...+\dfrac{1}{1001}\right)\)
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Tính: \(\left(\dfrac{1000}{1}+\dfrac{999}{2}+\dfrac{998}{3}+...+\dfrac{2}{999}+\dfrac{1}{1000}\right):\left(\dfrac{1}{2}+\dfrac{1}{3}+...+\dfrac{1}{1001}\right)\)
D=\(\left(1-\dfrac{1}{2}\right)\left(1-\dfrac{1}{3}\right)\left(1-\dfrac{1}{4}\right)...\left(1-\dfrac{1}{2005}\right)\)
E=\(\dfrac{1^2}{1.3}.\dfrac{2^2}{2.4}.\dfrac{3^2}{3.5}....\dfrac{999^2}{999.1000}.\dfrac{1000^2}{1000.1001}\)
Ta có: D\(=\left(1-\dfrac{1}{2}\right)\left(1-\dfrac{1}{3}\right)\left(1-\dfrac{1}{4}\right)...\left(1-\dfrac{1}{2005}\right)\)
\(\Leftrightarrow D=\dfrac{1}{2}.\dfrac{2}{3}.\dfrac{3}{4}...\dfrac{2004}{2005}=\dfrac{1.2.3...2004}{2.3.4...2005}=\dfrac{1}{2005}\)
Ta có: \(E=\dfrac{1^2}{1.3}.\dfrac{2^2}{2.4}.\dfrac{3^2}{3.5}...\dfrac{999^2}{999.1000}.\dfrac{1000^2}{1000.1001}=\dfrac{\left(1.2.3.4...1000\right)\left(1.2.3.4...1000\right)}{\left(1.2.3....1000\right)\left(3.4.5....1001\right)}=\dfrac{2}{1001}\)
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Tính A biết \(A=\dfrac{1000}{1}+\dfrac{999}{2}+\dfrac{998}{3}+...+\dfrac{2}{999}+\dfrac{1}{1000}\)
Yêu cầu bài toán chỉ đơn thuần tính cái này thôi à em!
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Bài 1: Tính tích
\(Q=\left(1-\dfrac{1}{2}\right)\left(1-\dfrac{1}{3}\right)\left(1-\dfrac{1}{4}\right)....\left(1-\dfrac{1}{999}\right)\left(1-\dfrac{1}{1000}\right)\)
Q=\(\dfrac{1}{2}\).\(\dfrac{2}{3}\).\(\dfrac{3}{4}\)...\(\dfrac{998}{999}\).\(\dfrac{999}{1000}\)
Q=\(\dfrac{1.2.3...998.999}{2.3.4....999.1000}\)
=>Q=\(\dfrac{1}{1000}\)
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Tính nhanh : \(\dfrac{1}{1}.\dfrac{1}{2}+\dfrac{1}{2}.\dfrac{1}{3}+\dfrac{1}{3}.\dfrac{1}{4}+...+\dfrac{1}{998}.\dfrac{1}{999}+\dfrac{1}{999}.\dfrac{1}{1000}\)
\(\dfrac{1}{1}.\dfrac{1}{2}+\dfrac{1}{2}.\dfrac{1}{3}+\dfrac{1}{3}.\dfrac{1}{4}+...+\dfrac{1}{999}.\dfrac{1}{1000}\\ =\dfrac{1}{1.2}+\dfrac{1}{2.3}+\dfrac{1}{3.4}+...+\dfrac{1}{999.1000}\\ =1-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{4}+...+\dfrac{1}{999}-\dfrac{1}{1000}\\ =1-\dfrac{1}{1000}=\dfrac{999}{1000}\)
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ta có
1/1.1/2=1-1/2
1/2.1/3=1/2-1/3
1/3.1/4=1/3-1/4
............
1/999.1/1000=1/999-1/1000
Từ đó suy ra
1/1.1/2+1/2-1/3+1/3+.......+1/998.1/999+1/999.1/1000
=1/1-1/2+1/2-1/3+1/3-.....+1/998-1/999+1/999-1/1000
=1-1/1000
=1000/1000-1/1000
=999/1000
nhớ like bạn nhé
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Tính A biết \(A=\dfrac{1000}{1}+\dfrac{999}{2}+\dfrac{998}{3}+...+\dfrac{2}{999}+\dfrac{1}{1000}\)
Tính nhanh :
Q = \(\left(\dfrac{1}{99}+\dfrac{12}{999}+\dfrac{123}{9999}\right).\left(\dfrac{1}{2}-\dfrac{1}{3}-\dfrac{1}{6}\right)\)
Ta có \(\dfrac{1}{2}-\dfrac{1}{3}-\dfrac{1}{6}=\dfrac{1}{6}-\dfrac{1}{6}=0\) nên Q = 0.
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\(Q=\left(\dfrac{1}{99}+\dfrac{12}{999}+\dfrac{123}{9999}\right).\left(\dfrac{1}{2}-\dfrac{1}{3}-\dfrac{1}{6}\right)\)
\(Q=\left(\dfrac{1}{99}+\dfrac{12}{999}+\dfrac{123}{9999}\right).0\)
\(Q=0\)
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Tính giá trị của biểu thức: \(Q=\left(\dfrac{1}{99}+\dfrac{12}{999}+\dfrac{123}{9999}\right)\left(\dfrac{1}{2}-\dfrac{1}{3}-\dfrac{1}{6}\right)\)
\(Q=\left(\dfrac{1}{99}+\dfrac{12}{999}+\dfrac{123}{9999}\right)\left(\dfrac{1}{2}-\dfrac{1}{3}-\dfrac{1}{6}\right)\)
\(Q=\left(\dfrac{1}{99}+\dfrac{12}{999}+\dfrac{123}{9999}\right)\left(\dfrac{3}{6}-\dfrac{2}{6}-\dfrac{1}{6}\right)\)
\(Q=\left(\dfrac{1}{99}+\dfrac{12}{999}+\dfrac{123}{9999}\right)\cdot\dfrac{0}{6}\)
\(Q=\left(\dfrac{1}{99}+\dfrac{12}{999}+\dfrac{123}{9999}\right)\cdot0\)
\(Q=0\)
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1, Tìm các số hữu tỉ:
a) Có dạng dfrac{12}{b} sao cho dfrac{-8}{19} dfrac{12}{b} dfrac{-2}{5}
b) Có dạng dfrac{9}{b} sao cho dfrac{8}{11} dfrac{9}{b} dfrac{12}{13}
2, Tính:
M54-dfrac{1}{2}left(1+2right)-dfrac{1}{3}left(1+2+3right)-dfrac{1}{4}left(1+2+3+4right)-...dfrac{1}{12}left(1+2+3+...+12right)
3, Rút gọn các biểu thức sau:
a) A dfrac{9^9+27^7}{9^6+243^3}
b) B dfrac{left(dfrac{2}{3}right)^5.left(dfrac{-27}{8}right)^2.729}{left(dfrac{3}{2}right)^4.216}
4, Cho a,b,c là các số nguyên dư...
Đọc tiếp
1, Tìm các số hữu tỉ:
a) Có dạng \(\dfrac{12}{b}\) sao cho \(\dfrac{-8}{19}< \dfrac{12}{b}< \dfrac{-2}{5}\)
b) Có dạng \(\dfrac{9}{b}\) sao cho \(\dfrac{8}{11}< \dfrac{9}{b}< \dfrac{12}{13}\)
2, Tính:
M=\(54-\dfrac{1}{2}\left(1+2\right)-\dfrac{1}{3}\left(1+2+3\right)-\dfrac{1}{4}\left(1+2+3+4\right)-...\dfrac{1}{12}\left(1+2+3+...+12\right)\)
3, Rút gọn các biểu thức sau:
a) A= \(\dfrac{9^9+27^7}{9^6+243^3}\)
b) B= \(\dfrac{\left(\dfrac{2}{3}\right)^5.\left(\dfrac{-27}{8}\right)^2.729}{\left(\dfrac{3}{2}\right)^4.216}\)
4, Cho a,b,c là các số nguyên dương sao cho mỗi số nhỏ hơn tổng của hai số kia. Chứng minh rằng \(\dfrac{a}{b+c}+\dfrac{b}{c+a}+\dfrac{c}{a+b}< 2\)
5, Cho A= \(\dfrac{1001}{1000^2+1}+\dfrac{1001}{1000^2+2}+...+\dfrac{1001}{1000^2+1000}\)
Chứng minh rằng 1<A2 < 4