\(C=\dfrac{9}{10}-\dfrac{1}{90}-\dfrac{1}{72}-\dfrac{1}{56}-\dfrac{1}{42}-\dfrac{1}{30}-\dfrac{1}{20}-\dfrac{1}{6}-\dfrac{1}{2}\)

\(\Leftrightarrow C=\dfrac{9}{10}-\left(\dfrac{1}{2}+\dfrac{1}{6}+\dfrac{1}{20}+\dfrac{1}{30}+\dfrac{1}{42}+\dfrac{1}{56}+\dfrac{1}{72}+\dfrac{1}{90}\right)\)

\(\Leftrightarrow C=\dfrac{9}{10}-\left(\dfrac{1}{1.2}+\dfrac{1}{2.3}+\dfrac{1}{3.4}+\dfrac{1}{4.5}+\dfrac{1}{5.6}+\dfrac{1}{7.8}+\dfrac{1}{9.10}\right)\)

\(\Leftrightarrow C=\dfrac{9}{10}-\left(1-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+.....+\dfrac{1}{9}-\dfrac{1}{10}\right)\)

\(\Leftrightarrow C=\dfrac{9}{10}-\left(1-\dfrac{1}{10}\right)\)

\(\Leftrightarrow C=\dfrac{9}{10}-\dfrac{9}{10}\)

\(\Leftrightarrow C=0\)

Bài 1:

\(A=2x+2y-y\)

\(A=2x+y\)

Thay x = 2,5 và y = 3/4 vào A

\(A=2.2,5+\dfrac{3}{4}\)

\(A=5+\dfrac{3}{4}\)

\(A=\dfrac{23}{4}\)

\(B=\dfrac{5a}{3}-\dfrac{3}{b}\)

Thay a = 1/3 và b = 0,25 vào B

\(B=\dfrac{5.\dfrac{1}{3}}{3}-\dfrac{3}{0,25}\)

\(B=\dfrac{5}{9}-12\)

\(B=-\dfrac{103}{9}\)

Bài 2:

a) \(\left(2x-\dfrac{1}{2}\right).2+\left(\dfrac{1}{2}+\dfrac{1}{3}+\dfrac{1}{4}\right):\dfrac{1}{8}=1\)

\(\Rightarrow4x-1+\dfrac{26}{3}=1\)

\(\Rightarrow4x+\dfrac{23}{3}=1\)

\(\Rightarrow4x=1-\dfrac{23}{3}\)

\(\Rightarrow4x=-\dfrac{20}{3}\)

\(\Rightarrow x=-\dfrac{5}{3}\)

b) \(\dfrac{x+1}{65}+\dfrac{x+3}{63}=\dfrac{x+5}{61}+\dfrac{x+7}{59}\)

\(\Rightarrow\dfrac{x+1}{65}+1+\dfrac{x+3}{63}+1=\dfrac{x+5}{61}+1+\dfrac{x+7}{59}+1\)

\(\Rightarrow\dfrac{x+66}{65}+\dfrac{x+66}{63}=\dfrac{x+66}{61}+\dfrac{x+66}{59}\)

\(\Rightarrow\left(x+66\right)\left(\dfrac{1}{65}+\dfrac{1}{63}\right)=\left(x+66\right)\left(\dfrac{1}{61}+\dfrac{1}{59}\right)\)

\(\Rightarrow\left(x+66\right)\left(\dfrac{1}{65}+\dfrac{1}{63}\right)-\left(x+66\right)\left(\dfrac{1}{61}+\dfrac{1}{59}\right)=0\)

\(\Rightarrow\left(x+66\right)\left(\dfrac{1}{65}+\dfrac{1}{63}-\dfrac{1}{61}-\dfrac{1}{59}\right)=0\)

Vì \(\dfrac{1}{65}+\dfrac{1}{63}-\dfrac{1}{61}-\dfrac{1}{59}\ne0\)

\(\Rightarrow x+66=0\)

\(\Rightarrow x=-66\)

Bài 3:

\(A=\left(1-\dfrac{1}{2}\right)\left(1-\dfrac{1}{3}\right)\left(1-\dfrac{1}{4}\right)...\left(1-\dfrac{1}{n}\right)\)

\(A=\dfrac{1}{2}.\dfrac{2}{3}.\dfrac{3}{4}...\dfrac{n-1}{n}\)

\(A=\dfrac{1}{n}\)

\(1+\dfrac{1}{2}\left(1+2\right)+\dfrac{1}{3}\left(1+2+3\right)+...+\dfrac{1}{n}\left(1+2+...+n\right)\)

\(=\dfrac{2}{2}+\dfrac{2.3}{2.2}+\dfrac{3.4}{3.2}+...+\dfrac{n\left(n+1\right)}{2n}\)

\(=\dfrac{2}{2}+\dfrac{3}{2}+\dfrac{4}{2}+...+\dfrac{n+1}{2}\)

\(=\dfrac{1}{2}\left(1+2+...+n\right)\)

\(=\dfrac{n\left(n+1\right)}{4}\)

P/s: \(1+...+n=\dfrac{n\left(n+1\right)}{2}\)