Bài 57:

\(A=\frac{2}{1\cdot4}+\frac{2}{4\cdot7}+\cdots+\frac{2}{97\cdot100}\)

\(=\frac23\left(\frac{3}{1\cdot4}+\frac{3}{4\cdot7}+\cdots+\frac{3}{97\cdot100}\right)\)

\(=\frac23\left(1-\frac14+\frac14-\frac17+\cdots+\frac{1}{97}-\frac{1}{100}\right)\)

\(=\frac23\left(1-\frac{1}{100}\right)=\frac23\cdot\frac{99}{100}=\frac{33}{50}\)

Bài 52:

\(A=\frac{4}{7\cdot31}+\frac{6}{7\cdot41}+\frac{9}{10\cdot41}+\frac{7}{10\cdot57}\)

\(=\frac{20}{31\cdot35}+\frac{30}{35\cdot41}+\frac{45}{41\cdot50}+\frac{35}{50\cdot57}\)

\(=5\left(\frac{4}{31\cdot35}+\frac{6}{35\cdot41}+\frac{9}{41\cdot50}+\frac{7}{50\cdot57}\right)\)

\(=5\left(\frac{1}{31}-\frac{1}{35}+\frac{1}{35}-\frac{1}{41}+\frac{1}{41}-\frac{1}{50}+\frac{1}{50}-\frac{1}{57}\right)\)

\(=5\left(\frac{1}{31}-\frac{1}{57}\right)\)

\(B=\frac{7}{19\cdot31}+\frac{5}{19\cdot43}+\frac{3}{23\cdot43}+\frac{11}{23\cdot57}\)

\(=\frac{14}{31\cdot38}+\frac{10}{38\cdot43}+\frac{6}{43\cdot46}+\frac{22}{46\cdot57}\)

\(=2\left(\frac{7}{31\cdot38}+\frac{5}{38\cdot43}+\frac{3}{43\cdot46}+\frac{11}{46\cdot57}\right)\)

\(=2\left(\frac{1}{31}-\frac{1}{38}+\frac{1}{38}-\frac{1}{43}+\frac{1}{43}-\frac{1}{46}+\frac{1}{46}-\frac{1}{57}\right)=2\left(\frac{1}{31}-\frac{1}{57}\right)\)

Do đó: \(\frac{A}{B}=\frac52\)

Bài 53:

\(A=\frac{34}{7\cdot13}+\frac{51}{13\cdot22}+\frac{85}{22\cdot37}+\frac{68}{37\cdot49}\)

\(=17\cdot\left(\frac{2}{7\cdot13}+\frac{3}{13\cdot22}+\frac{5}{22\cdot37}+\frac{4}{37\cdot49}\right)\)

\(=\frac{17}{3}\left(\frac{6}{7\cdot13}+\frac{9}{13\cdot22}+\frac{15}{22\cdot37}+\frac{12}{37\cdot49}\right)\)

\(=\frac{17}{3}\left(\frac17-\frac{1}{13}+\frac{1}{13}-\frac{1}{22}+\frac{1}{22}-\frac{1}{37}+\frac{1}{37}-\frac{1}{49}\right)=\frac{17}{3}\left(\frac17-\frac{1}{49}\right)\)

\(B=\frac{39}{7\cdot16}+\frac{65}{16\cdot31}+\frac{52}{31\cdot43}+\frac{26}{43\cdot49}\)

\(=13\left(\frac{3}{7\cdot16}+\frac{5}{16\cdot31}+\frac{4}{31\cdot43}+\frac{2}{43\cdot49}\right)\)

\(=\frac{13}{3}\left(\frac{9}{7\cdot16}+\frac{15}{16\cdot31}+\frac{12}{31\cdot43}+\frac{6}{43\cdot49}\right)\)

\(=\frac{13}{3}\left(\frac17-\frac{1}{16}+\frac{1}{16}-\frac{1}{31}+\frac{1}{31}-\frac{1}{43}+\frac{1}{43}-\frac{1}{49}\right)=\frac{13}{3}\left(\frac17-\frac{1}{49}\right)\)

Do đó: \(\frac{A}{B}=\frac{17}{3}:\frac{13}{3}=\frac{17}{13}\)