D=10/56 + 10/140 + 10/260 +...+ 10/1400 ; E=1/1+2 + 1/1+2+3 + 1/1+2+3+4 +...+ 1/1+2+3+...+24 .Tính tỉ số của D/E
$* \ D=\dfrac{10}{56}+\dfrac{10}{140}+\dfrac{10}{260}+...+\dfrac{10}{1400}
\\=\dfrac{5}{28}+\dfrac{5}{70}+\dfrac{5}{130}+...+\dfrac{5}{280}
\\=\dfrac{5}{3}(\dfrac{3}{4.7}+\dfrac{3}{7.10}+\dfrac{3}{10.13}+...+\dfrac{3}{25.28})
\\=\dfrac{5}{3}(\dfrac14-\dfrac17+\dfrac17-\dfrac1{10}+\dfrac1{10}-\dfrac1{13}+...+\dfrac1{25}-\dfrac1{28})
\\=\dfrac 53(\dfrac14-\dfrac1{28})=\dfrac 53.\dfrac 3{14}=\dfrac 5{14}$
$ * \ E=\dfrac1{1+2}+\dfrac1{1+2+3}+\dfrac1{1+2+3+4}+...+\dfrac1{1+2+3+...+24}
\\=\dfrac1{\dfrac{(1+2).2}2}+\dfrac1{\dfrac{(1+3).3}2}+\dfrac1{\dfrac{(1+4).4}2}+...+\dfrac1{\dfrac{(1+24).24}2}
\\=2(\dfrac 1{2.3}+\dfrac 1{3.4}+\dfrac 1{4.5}+...+\dfrac 1{24.25})
\\=2(\dfrac12-\dfrac13+\dfrac13-\dfrac14+\dfrac14-\dfrac15+...+\dfrac1{24}-\dfrac1{25})
\\=2(\dfrac12-\dfrac1{25})=2.\dfrac{23}{50}=\dfrac{23}{25}$
$\Rightarrow \dfrac DE=\dfrac 5{14}:\dfrac{23}{25}=\dfrac{125}{322}$
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