\(\dfrac{1+3+6+10+...+45+55}{1x10+9x2+3x8+...9x2+10x1}\)
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B
0
NL
Nguyễn Lê Phước Thịnh
CTVHS
8 tháng 11 2025
Ta có: \(1\times10+2\times9+\cdots+10\times1\)
\(=2\times\left(1\times10+2\times9+\cdots+5\times6\right)\)
\(=2\times\left(11+18+24+28+30\right)=2\times111=222\)
\(1+3+6+\cdots+45+55\)
\(=\frac{1\times2}{2}+\frac{2\times3}{2}+\frac{3\times4}{2}+\cdots+\frac{9\times10}{2}+\frac{10\times11}{2}\)
\(=\frac12\times\left(1\times2+2\times3+\cdots+10\times11\right)\)
\(=\frac12\times\left\lbrack1\times\left(1+1\right)+2\times\left(2+1\right)+\cdots+10\times\left(10+1\right)\right\rbrack\)
\(=\frac12\times\left\lbrack\left(1\times1+2\times2+\cdots+10\times10\right)+\left(1+2+\cdots+10\right)\right\rbrack\)
\(=\frac12\times\left\lbrack\frac{10\times\left(10+1\right)\times\left(2\times10+1\right)}{6}+\frac{10\times11}{2}\right\rbrack\)
\(=\frac12\times\left\lbrack\frac{10\times11\times21}{6}+5\times11\right\rbrack=\frac12\times\left\lbrack5\times11\times7+5\times11\right\rbrack\)
\(=\frac12\times55\times\left(7+1\right)=55\times\frac82=55\times4=220\)
Ta có: \(\frac{1+3+6+\cdots+45+55}{1\times10+2\times9+\cdots+10\times1}\)
\(=\frac{220}{222}=\frac{110}{111}\)
NL
Nguyễn Lê Phước Thịnh
CTVHS
21 tháng 11 2025
Ta có: \(\frac{1}{1\times10}+\frac{1}{2\times9}+\cdots+\frac{1}{10\times1}\)
\(=2\times\left(\frac{1}{1\times10}+\frac{1}{2\times9}+\cdots+\frac{1}{5\times6}\right)\)
\(=\frac{2}{11}\times\left(\frac{11}{1\times10}+\frac{11}{2\times9}+\cdots+\frac{11}{5\times6}\right)\)
\(=\frac{2}{11}\times\left(1+\frac12+\frac13+\cdots+\frac19+\frac{1}{10}\right)\)
Ta có: \(A=\frac{1+\frac12+\cdots+\frac{1}{10}}{\frac{1}{1\times10}+\frac{1}{2\times9}+\cdots+\frac{1}{9\times2}+\frac{1}{10\times1}}\)
\(=\frac{1+\frac12+\cdots+\frac{1}{10}}{\frac{2}{11}\times\left(1+\frac12+\cdots+\frac{1}{10}\right)}=1:\frac{2}{11}=\frac{11}{2}\)
LH
0
NL
Nguyễn Lê Phước Thịnh
CTVHS
21 tháng 11 2025
Ta có: \(\frac{1}{1\times10}+\frac{1}{2\times9}+\cdots+\frac{1}{10\times1}\)
\(=2\times\left(\frac{1}{1\times10}+\frac{1}{2\times9}+\cdots+\frac{1}{5\times6}\right)\)
\(=\frac{2}{11}\times\left(\frac{11}{1\times10}+\frac{11}{2\times9}+\cdots+\frac{11}{5\times6}\right)\)
\(=\frac{2}{11}\times\left(1+\frac12+\frac13+\cdots+\frac19+\frac{1}{10}\right)\)
Ta có: \(A=\frac{1+\frac12+\cdots+\frac{1}{10}}{\frac{1}{1\times10}+\frac{1}{2\times9}+\cdots+\frac{1}{9\times2}+\frac{1}{10\times1}}\)
\(=\frac{1+\frac12+\cdots+\frac{1}{10}}{\frac{2}{11}\times\left(1+\frac12+\cdots+\frac{1}{10}\right)}=1:\frac{2}{11}=\frac{11}{2}\)
Ta có: \(1+3+6+10+\cdots+45+55\)
\(=\frac12\times\left(2+6+12+20+\cdots+90+110\right)\)
\(=\frac12\times\left(1\times2+2\times3+\cdots+9\times10+10\times11\right)\)
\(=\frac12\times\left\lbrack1\times\left(1+1\right)+2\times\left(2+1\right)+\cdots+10\times\left(10+1\right)\right\rbrack\)
\(=\frac12\times\left\lbrack\left(1\times1+2\times2+\cdots+10\times10\right)+\left(1+2+\cdots+10\right)\right\rbrack\)
\(=\frac12\times\left\lbrack10\times\left(10+1\right)\times\frac{\left(2\times10+1\right)}{6}+10\times\frac{11}{2}\right\rbrack\)
\(=\frac12\times\left\lbrack10\times11\times\frac{21}{6}+5\times11\right\rbrack=\frac12\times\left\lbrack5\times11\times7+5\times11\right\rbrack\)
\(=\frac12\times5\times11\times\left(7+1\right)=\frac{55}{2}\times8=55\times4=220\)
Ta có: \(1\times10+9\times2+\cdots+10\times1\)
\(=2\times\left(1\times10+2\times9+3\times8+4\times7+5\times6\right)\)
\(=2\times\left\lbrack1\times\left(11-1\right)+2\times\left(11-2\right)+3\times\left(11-3\right)+4\times\left(11-4\right)+5\times\left(11-5\right)\right\rbrack\)
\(=2\times\left\lbrack11\times\left(1+2+3+4+5\right)-\left(1\times1+2\times2+3\times3+4\times4+5\times5\right)\right\rbrack\)
\(=2\times\left\lbrack11\times15-\left(1+4+9+16+25\right)\right\rbrack\)
=2x(165-55)
=2x110
=220
Ta có: \(\frac{1\times10+9\times2+\cdots+10\times1}{1+3+6+10+\cdots+45+55}\)
\(=\frac{220}{220}\)
=1