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Ta có: \(1\times2010+2\times2009+\cdots+2010\times1\)
\(=2\times\left(1\times2010+2\times2009+\cdots+1004\times1007+1005\times1006\right)\)
\(=2\times\left\lbrack1\times\left(2011-1\right)+2\times\left(2011-2\right)+\cdots+1005\times\left(2011-1005\right)\right\rbrack\)
\(=2\times\left\lbrack2011\times\left(1+2+\cdots+1005\right)-\left(1\times1+2\times2+\cdots+1005\times1005\right)\right\rbrack\)
\(=2\times\left\lbrack2011\times1005\times\frac{1006}{2}-\frac{1005\times\left(1005+1\right)\times\left(2\times1005+1\right)}{6}\right\rbrack\)
\(=2\times\left\lbrack2011\times1005\times503-\frac{1005\times1006\times2011}{6}\right\rbrack\)
\(=2\times\left\lbrack2011\times1005\times503-335\times503\times2011\right\rbrack\)
=2x2011x335x503x(3-1)
=2x2011x335x503x2
=2011x2012x335
Ta có: \(\left(1+2+3+\cdots+2010\right)+\left(1+2+\cdots+2009\right)+\cdots+\left(1+2\right)+1\)
\(=\frac{2010\times2011}{2}+\frac{2009\times2010}{2}+\cdots+\frac{2\times3}{2}+\frac{1\times2}{2}\)
\(=\frac12\times\left(1\times2+2\times3+\cdots+2010\times2011\right)\)
\(=\frac12\times\left\lbrack1\times\left(1+1\right)+2\times\left(2+1\right)+\cdots+2010\times\left(2010+1\right)\right\rbrack\)
\(=\frac12\times\left\lbrack\left(1\times1+2\times2+\cdots+2010\times2010\right)+\left(1+2+\cdots+2010\right)\right\rbrack\)
\(=\frac12\times\left\lbrack\frac{2010\times\left(2010+1\right)\times\left(2\times2010+1\right)}{6}+\frac{2010\times2011}{2}\right\rbrack\)
\(=\frac12\times\left\lbrack335\times2011\times4021+1005\times2011\right\rbrack\)
\(=\frac12\times\left\lbrack2011\times335\times\left(4021+3\right)\right\rbrack=\frac12\times\left\lbrack2011\times335\times4024\right\rbrack\)
=2011x335x2012
Ta có: \(A=\frac{1\times2010+2\times2009+\cdots+2010\times1}{\left(1+2+3+\cdots+2010\right)+\left(1+2+\cdots+2009\right)+\cdots+\left(1+2\right)+1}\)
\(=\frac{2011\times2012\times335}{2011\times2012\times335}=1\)
A) 2 và 1/3 ÷7/9+2/5× 1 và 2/3 -7/3
= 7/3 : 7/9 + 2/5 x 5/3 - 7/3
= 7/3 x 9/7 + 2/5 x 5/3 - 7/3
= 7/3 x ( 9/7 + 2/5 x 5/3 )
= 7/3 x ( 9/7 + 2/3 )
= 7/3 x 41/21= 41/9
B) 25% ×4/5 +3/5 -0.8 +2010
= (1/4 ×4/5 ) + ( 3/5 - 4/5 ) +2010
= 1/5 + -1/5 + 2010
= 0 + 2010
= 2010
Ta có: \(A=1\times2010+2\times2009+\cdots+2010\times1\)
\(=2\times\left(1\times2010+2\times2009+\cdots+1005\times1006\right)\)
\(=2\times\left\lbrack1\times\left(2011-1\right)+2\times\left(2011-2\right)+\cdots+1005\times\left(2011-1005\right)\right\rbrack\)
\(=2\times\left\lbrack2011\left(1+2+\cdots+1005\right)-\left(1\times1+2\times2+\ldots+1005\times1005\right)\right\rbrack\)
\(=2\times\left\lbrack2011\times1005\times\frac{1006}{2}-\frac{1005\times\left(1005+1\right)\times\left(2\times1005+1\right)}{6}\right\rbrack\)
\(=2\left\lbrack2011\times1005\times\frac{1006}{2}-\frac{1005\times1006\times2011}{6}\right\rbrack\)
\(=2011\times1005\times1006-\frac{1005\times1006\times2011}{3}\)
\(=2011\times1005\times1006-335\times1006\times2011=2011\times1006\times335\left(3-1\right)=2\times335\times1006\times2011\)
=335x2011x2012
Ta có: B=1+(1+2)+(1+2+3)+...+(1+2+3+...+2010)
\(=\frac{1\times2}{2}+\frac{2\times3}{2}+\cdots+\frac{2010\times2011}{2}\)
\(=\frac12\times\left(1\times2+2\times3+\cdots+2010\times2011\right)\)
\(=\frac12\times\left\lbrack1\times\left(1+1\right)+\right.2\times\left(2+1)+\cdots+2010\times\left(2010+1\right)\right\rbrack\)
\(=\frac12\times\left\lbrack\left(1\times1+2\times2+\cdots+2010\times2010\right)+\left(1+2+\cdots+2010\right)\right\rbrack\)
\(=\frac12\times\left\lbrack\frac{2010\times\left(2010+1\right)\left(2\times2010+1\right)}{6}+\frac{2010\times2011}{2}\right\rbrack\)
\(=\frac12\times\left\lbrack\frac{2010\times2011\times4021}{6}+\frac{2010\times2011}{2}\right\rbrack=\frac12\times\frac{2010\times2011\times4021+3\times2010\times2011}{6}\)
\(=\frac12\times\frac16\times2010\times2011\times\left(4021+3\right)=\frac12\times\frac16\times2010\times2011\times4024\)
\(=2010\times2011\times\frac{2012}{6}=2011\times2012\times335\)
=A
=>A:B=1