\(\frac{2,4x1994x2+1,6x3996x3+1,2x4010x4}{3+7+11+15+...+95+99-275}\)
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1. 2,4 x 1994 x 2 + 1,6 x 3996 x 3 + 1,2 x 4010 x 4
= ( 2,4 x 1994 x 2 ) + ( 1,6 x 3996 x 3 ) + ( 1,2 x 4010 x 4 )
= ( 2,4 x 2 x 1994 ) + ( 1,6 x 3 x 3996 ) + ( 1,2 x 4 x 4010 )
= 4,8 x 1994 + 4,8 x 3996 + 4,8 x 4010
= 4,8 x ( 1994 + 3996 + 4010 )
= 4,8 x 10 000
= 48000
2. 3 + 7 + 11 + 15 +........+ 95 + 99 - 275
= [ ( 99 - 3 ) : 4 + 1 ] - 275
= 25 - 275
= [ ( 99 + 3 ) x 25 : 2 ] - 275
= 1275 - 275
= 1000
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\(\frac{2,4\times1994\times2+1,6\times3996\times3+1,2\times4010\times4}{3+7+11+...+95+99-275}\)
\(=\frac{0,2\times6\times2\times2\times997\times2+0,2\times8\times6\times666\times3+0,2\times6\times2\times2005\times4}{\left(99+3\right)\times49:2-275}\)
\(=\frac{0,2\times6\times8\times997+0,2\times8\times6\times666\times3+0,2\times6\times8\times2005}{2499-275}\)
\(=\frac{0,2\times6\times8\times\left(997+666\times3+2005\right)}{2224}\)
\(=\frac{0,2\times2\times3\times8\times\left(997+1998+2005\right)}{8\times2\times0,2\times695}\)
\(=\frac{3\times\left(2995+2005\right)}{695}\)
\(=\frac{3\times5000}{695}=\frac{3\times1000\times5}{5\times139}\)
\(=\frac{3\times1000}{139}=\frac{3000}{139}\)
Ta có: \(A=\frac{1}{3}.\frac{1}{7}+\frac{1}{7}.\frac{1}{11}+\frac{1}{11}.\frac{1}{15}+...+\frac{1}{95}.\frac{1}{99}\)
\(=\frac{1}{4}.\left(\frac{1}{3}-\frac{1}{7}+\frac{1}{7}-\frac{1}{11}+.....+\frac{1}{95}-\frac{1}{99}\right)\)
\(=\frac{1}{4}.\left(\frac{1}{3}-\frac{1}{99}\right)\)
\(=\frac{1}{4}.\frac{32}{99}=\frac{8}{99}\)
\(4A=\frac{4}{3.7}+...+\frac{4}{95.99}\)
\(4A=\frac{1}{3}-\frac{1}{7}+\frac{1}{7}-\frac{1}{11}+...+\frac{1}{95}-\frac{1}{99}\)
\(4A=\frac{1}{3}-\frac{1}{99}=\frac{32}{99}\)
\(\Rightarrow A=\frac{8}{99}\)
\(A=\frac{1}{3.7}+\frac{1}{7.11}+...+\frac{1}{95.99}\)
\(A=\frac{1}{4}.\left(\frac{4}{3.7}+\frac{4}{7.11}+...+\frac{4}{95.99}\right)\)
\(A=\frac{1}{4}.\left(\frac{1}{3}-\frac{1}{7}+\frac{1}{7}-\frac{1}{11}+...+\frac{1}{95}-\frac{1}{99}\right)\)
\(A=\frac{1}{4}.\left(\frac{1}{3}-\frac{1}{99}\right)\)
\(A=\frac{1}{4}.\frac{32}{99}\)
\(A=\frac{8}{99}\)