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Ta có công thức : \(1+2+3+...+n=\frac{n.\left(n+1\right)}{2}\)
\(\Rightarrow B=\frac{1}{1+2}+\frac{1}{1+2+3}+...+\frac{1}{1+2+...+10}\)
\(=\frac{1}{\frac{\left(1+2\right).2}{2}}+\frac{1}{\frac{\left(1+3\right).3}{2}}+...+\frac{1}{\frac{\left(1+10\right)10}{2}}\)
\(=\frac{2}{2.3}+\frac{2}{3.4}+...+\frac{2}{10.11}\)
\(=2.\left(\frac{1}{2.3}+\frac{1}{3.4}+...+\frac{1}{10.11}\right)\)
\(=2.\left(\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{10}-\frac{1}{11}\right)\)
\(=2.\left(\frac{1}{2}-\frac{1}{11}\right)=2.\frac{9}{22}=\frac{9}{11}\)
\(\frac{4.\left(\frac{1}{5}-\frac{1}{71}+\frac{1}{123}\right)}{3.\left(\frac{1}{5}-\frac{1}{71}+\frac{1}{123}\right)}-\frac{\frac{1}{19}+\frac{1}{1315}-\frac{1}{287}}{3.\left(\frac{1}{19}+\frac{1}{1315}-\frac{1}{287}\right)}=\frac{4}{3}-\frac{1}{3}=\frac{3}{3}=1\)
Ta có:
\(1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{70}=\left[1+\frac{1}{70}\right]+\left[\frac{1}{2}+\frac{1}{69}\right]+\left[\frac{1}{3}+\frac{1}{68}\right]+...+\left[\frac{1}{35}+\frac{1}{36}\right]\)
\(=\frac{71}{1.70}+\frac{71}{2.69}+\frac{71}{3.68}+...+\frac{71}{35.36}\)
\(=71\left[\frac{1}{1.70}+\frac{1}{2.69}+\frac{1}{3.68}+...+\frac{1}{35.36}\right]⋮71\)
=> \(A=1\times2\times3\times4\times...\times70\times\left[1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{70}\right]⋮71\)=> ĐPCM
AI THẤY ĐÚNG NHỚ ỦNG HỘ NHA
Xét \(1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{70}=\left(1+\frac{1}{70}\right)+\left(\frac{1}{2}+\frac{1}{69}\right)+...+\left(\frac{1}{35}+\frac{1}{36}\right)\)
\(=\frac{71}{1.70}+\frac{71}{2.69}+...+\frac{71}{35.36}=71\left(\frac{1}{1.70}+\frac{1}{2.69}+...+\frac{1}{35.36}\right)\)
=>\(A=1.2.3.4...71.\left(\frac{1}{1.70}+\frac{1}{2.69}+...+\frac{1}{35.36}\right)⋮71\)
Vậy A chia hết cho 71
\(A=\frac{1}{2}.\frac{2}{3}.\frac{3}{4}...\frac{99}{100}\)
\(A=\frac{1.2.3...99}{2.3.4...100}\)
\(A=\frac{1}{100}\)
\(B=1-\frac{1}{12}+1-\frac{1}{20}+...+1-\frac{1}{72}\)
\(B=1+1+...+1+\left(\frac{1}{12}+\frac{1}{20}+...+\frac{1}{72}\right)\)
\(B=5.1+\left(\frac{1}{3.4}+\frac{1}{4.5}+...+\frac{1}{8.9}\right)\)
\(B=5+\left(\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+...+\frac{1}{8}-\frac{1}{9}\right)\)
\(B=5+\left(\frac{1}{3}-\frac{1}{9}\right)\)
\(B=5+\frac{2}{9}=\frac{47}{9}\)
\(A=\left(1-\frac{1}{2}\right)\left(1-\frac{1}{3}\right)\left(1-\frac{1}{4}\right)......\left(1-\frac{1}{100}\right)\)
\(=\frac{1}{2}.\frac{2}{3}.\frac{3}{4}.....\frac{99}{100}\)
\(=\frac{1.2.3.4....99}{2.3.4.5...100}\)
\(=\frac{1}{100}\)
\(A=\frac{1}{2}+\frac{5}{6}+\frac{11}{12}+\frac{19}{20}+\frac{29}{30}+\frac{41}{42}+\frac{55}{56}+\frac{71}{72}+\frac{89}{90}\)
\(A=\left(1-\frac{1}{2}\right)+\left(1-\frac{1}{6}\right)+...+\left(1-\frac{1}{90}\right)\)
\(A=1+1+...+1-\left(\frac{1}{2}+\frac{1}{6}+...+\frac{1}{90}\right)\)
\(A=9-\left(\frac{1}{1.2}+\frac{1}{2.3}+...+\frac{1}{9.10}\right)=9-\left(\frac{1}{1}-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+...+\frac{1}{9}-\frac{1}{10}\right)\)
\(A=9-\left(1-\frac{1}{10}\right)=9-1+\frac{1}{10}=8\frac{1}{10}\)
\(A=1-\frac{1}{2}+1-\frac{1}{6}+1-\frac{1}{12}+1-\frac{1}{20}+1-\frac{1}{30}+1-\frac{1}{42}+1-\frac{1}{56}+1-\frac{1}{72}+1-\frac{1}{90}\)
=>\(A=\left(1+1+1+1+1+1+1+1+1\right)-\left(\frac{1}{2}+\frac{1}{6}+\frac{1}{12}+\frac{1}{20}+\frac{1}{30}+\frac{1}{42}+\frac{1}{56}+\frac{1}{72}+\frac{1}{90}\right)\)
=>\(A=9-\left(\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+\frac{1}{4.5}+\frac{1}{5.6}+\frac{1}{6.7}+\frac{1}{7.8}+\frac{1}{8.9}+\frac{1}{9.10}\right)\)
=>\(A=9-\left(1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+\frac{1}{5}-\frac{1}{6}+\frac{1}{6}-\frac{1}{7}+\frac{1}{7}-\frac{1}{8}+\frac{1}{8}-\frac{1}{9}+\frac{1}{9}-\frac{1}{10}\right)\)
=>\(a=9-\left(1-\frac{1}{10}\right)=\frac{90}{10}-\frac{9}{10}=\frac{81}{10}\)
9 - A = \(1-\frac{1}{2}+1-\frac{5}{6}+1-\frac{11}{12}+..+1-\frac{89}{90}=\frac{1}{2}+\frac{1}{6}+..+\frac{1}{90}\)
= \(\frac{1}{1.2}+\frac{1}{2.3}+...+\frac{1}{9.10}=\frac{1}{1}-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+...+\frac{1}{9}-\frac{1}{10}=\frac{1}{1}-\frac{1}{10}=\frac{9}{10}\)
=> A = \(9-\frac{9}{10}=\frac{81}{10}\)
=> A = ( 1 - 1/2 ) + ( 1 - 1/6 ) + ( 1 - 1/12 ) + ( 1 - 1/30 ) + .... + ( 1 - 1/90 )
=> A = ( 1 + 1 + 1 + .... 1 ) - ( 1/2 + 1/6 + 1/12 + 1/30 + .... + 1/90 )
=> A = 9 - ( 1/1.2 + 1/2.3 + 1/3.4 + 1/4.5 + .... + 1/9.10 )
=> A = 9 - ( 1 - 1/2 + 1/2 - 1/3 + 1/3 - 1/4 + 1/4 - 1/5 + ... + 1/9 - 1/10 )
=> A = 9 - ( 1 - 1/10 )
=> A = 9 - 9/10
=> 81/10