5.491 Найдите значение выражения: а) $$\frac{61}{64} - (\frac{7}{12} - \frac{5}{14}) \cdot (\frac{13}{16} + \frac{1}{2})$$; б) $$(1 - \frac{11}{17}) \cdot (\frac{3}{4} - \frac{5}{12} + \frac{11}{18})$$; в) $$1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \frac{1}{5} - \frac{1}{6}$$; г) $$\frac{1}{8} + \frac{3}{8} + \frac{1}{12} + \frac{5}{12} + \frac{1}{18} + \frac{7}{18} + ... + \frac{1}{99} + \frac{197}{99}$$

а) $$\frac{61}{64} - (\frac{7}{12} - \frac{5}{14}) \cdot (\frac{13}{16} + \frac{1}{2})$$ $$\frac{7}{12} - \frac{5}{14} = \frac{7 \cdot 7}{12 \cdot 7} - \frac{5 \cdot 6}{14 \cdot 6} = \frac{49}{84} - \frac{30}{84} = \frac{19}{84}$$ $$\frac{13}{16} + \frac{1}{2} = \frac{13}{16} + \frac{1 \cdot 8}{2 \cdot 8} = \frac{13}{16} + \frac{8}{16} = \frac{21}{16}$$ $$\frac{19}{84} \cdot \frac{21}{16} = \frac{19 \cdot 21}{84 \cdot 16} = \frac{399}{1344} = \frac{19}{64}$$ $$\frac{61}{64} - \frac{19}{64} = \frac{61-19}{64} = \frac{42}{64} = \frac{21}{32}$$ б) $$(1 - \frac{11}{17}) \cdot (\frac{3}{4} - \frac{5}{12} + \frac{11}{18})$$ $$1 - \frac{11}{17} = \frac{17}{17} - \frac{11}{17} = \frac{6}{17}$$ $$\frac{3}{4} - \frac{5}{12} + \frac{11}{18} = \frac{3 \cdot 9}{4 \cdot 9} - \frac{5 \cdot 3}{12 \cdot 3} + \frac{11 \cdot 2}{18 \cdot 2} = \frac{27}{36} - \frac{15}{36} + \frac{22}{36} = \frac{27-15+22}{36} = \frac{34}{36} = \frac{17}{18}$$ $$\frac{6}{17} \cdot \frac{17}{18} = \frac{6 \cdot 17}{17 \cdot 18} = \frac{6}{18} = \frac{1}{3}$$ в) $$1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \frac{1}{5} - \frac{1}{6} = \frac{60-30+20-15+12-10}{60} = \frac{37}{60}$$ г) $$\frac{1}{8} + \frac{3}{8} + \frac{1}{12} + \frac{5}{12} + \frac{1}{18} + \frac{7}{18} + ... + \frac{1}{99} + \frac{197}{99} = (\frac{1}{8} + \frac{3}{8}) + (\frac{1}{12} + \frac{5}{12}) + (\frac{1}{18} + \frac{7}{18}) + ... + (\frac{1}{99} + \frac{197}{99}) = \frac{4}{8} + \frac{6}{12} + \frac{8}{18} + ... + \frac{198}{99} = \frac{1}{2} + \frac{1}{2} + \frac{4}{9} + ... + 2 = 1 + \frac{4}{9} + ... + 2$$ К сожалению, без большего количества членов последовательности сложно точно определить закономерность и сумму.