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Bài 1:
\(\frac{1}{2}.\frac{1}{3}+\frac{1}{3}.\frac{1}{4}+....+\frac{1}{8}.\frac{1}{9}\)
\(=\frac{1}{2.3}+\frac{1}{3.4}+....+\frac{1}{8.9}\)
\(=\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+.....+\frac{1}{8}-\frac{1}{9}\)
\(=\frac{1}{2}-\frac{1}{9}=\frac{7}{18}\)
\(1+\frac{1}{3}+\frac{1}{6}+\frac{1}{10}+...+\frac{1}{x\left(x+1\right):2}=1\frac{2016}{2017}\)
\(\Rightarrow\frac{1}{2}\left(1+\frac{1}{3}+\frac{1}{6}+\frac{1}{10}+...+\frac{2}{x\left(x+1\right)}\right)=\frac{1}{2}.\frac{4033}{2017}\)
\(\Leftrightarrow\frac{1}{2}+\frac{1}{6}+\frac{1}{12}+\frac{1}{20}+...+\frac{1}{x\left(x+1\right)}=\frac{4033}{4034}\)
\(\Rightarrow\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+\frac{1}{4.5}+...+\frac{1}{x\left(x+1\right)}=\frac{4033}{4034}\)
\(\Rightarrow1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+...+\frac{1}{x}-\frac{1}{x+1}=\frac{4033}{4034}\)
\(\Rightarrow1-\frac{1}{x+1}=\frac{4033}{4034}\)
\(\Rightarrow\frac{1}{x+1}=1-\frac{4033}{4034}\)
\(\Rightarrow\frac{1}{x+1}=\frac{1}{4034}\)
\(\Rightarrow x+1=4034\)
\(\Rightarrow x=4034-1\)
\(\Rightarrow x=4033\)
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Câu hỏi của Best Friend Forever - Toán lớp 7 - Học toán với OnlineMath
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\(\frac{2.2}{1.3}.\frac{3.3}{2.4}.\frac{4.4}{3.5}......\frac{100.100}{99.101}=\frac{2.3.4...100}{1.2.3...99}.\frac{2.3.4...100}{3.4.5...101}=100.\frac{2}{101}=\frac{200}{101}\)
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Xét thừa số tổng quát:
\(\frac{1+2+...+n}{n}=\frac{n\left(n+1\right):2}{n}=\frac{n+1}{2}\)
Thay vào bài toán:
\(E=1+\frac{1}{2}\left(1+2\right)+\frac{1}{3}\left(1+2+3\right)+...+\frac{1}{200}\left(1+2+3+...+200\right)\)
\(E=1+\frac{1+2}{2}+\frac{1+2+3}{3}+...+\frac{1+2+3+...+200}{200}\)
\(E=1+\frac{2+1}{2}+\frac{3+1}{2}+...+\frac{200+1}{2}\)
\(E=\frac{2}{2}+\frac{3}{2}+\frac{4}{2}+...+\frac{201}{2}\)
\(E=\frac{2+3+4+...+201}{2}=\frac{20300}{2}=10150\)
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Sai đâu bỏ qua nhé, hơi to mới lại mk tính máy tính ra : \(\frac{77}{30}\)nên ko chắc nhé
\(2+\frac{1}{1+\frac{1}{1+\frac{1}{3+\frac{1}{4}}}}=2+\frac{1}{1+\frac{1}{1+\frac{1}{\frac{13}{4}}}}\)
\(=2+\frac{1}{1+\frac{1}{1+\frac{4}{13}}}=2+\frac{1}{1+\frac{1}{\frac{17}{3}}}\)
\(=2+\frac{1}{1+\frac{3}{17}}=2+\frac{1}{\frac{20}{17}}=2+\frac{17}{20}=\frac{57}{20}\)
(1-1/2).(1.1/3).(1.1-4)....(1-1/2016)
=1/2.2/3.3/4.4/5....2015/2016
=1.2.3....2015/2.3.4....2016
=1/2016
1/2016