Odpowiedź:
a) [tex]10[/tex]
b) [tex]7[/tex]
c) [tex]673[/tex]
d) [tex]1010[/tex]
Szczegółowe wyjaśnienie:
a)
[tex](1+\frac{1}{2})+(\frac{1}{2}+\frac{1}{3})+(\frac{2}{3}+\frac{1}{4})+(\frac{3}{4}+\frac{1}{5})+(\frac{4}{5}+\frac{1}{6})+(\frac{5}{6}+\frac{1}{7})+(\frac{6}{7}+\frac{1}{8})+(\frac{7}{8}+\frac{1}{9})+(\frac{8}{9}+\frac{1}{10})+\frac{9}{10}=\\\\=1+(\frac{1}{2}+\frac{1}{2})+(\frac{1}{3}+\frac{2}{3})+(\frac{1}{4}+\frac{3}{4})+(\frac{1}{5}+\frac{4}{5})+(\frac{1}{6}+\frac{5}{6})+(\frac{1}{7}+\frac{6}{7})+(\frac{1}{8}+\frac{7}{8})+(\frac{1}{9}+\frac{8}{9})+(\frac{1}{10}+\frac{9}{10})=\\\\\\=1\cdot 10=10[/tex]
b)
[tex](1+\frac{2}{3})\cdot(1+\frac{2}{5})\cdot(1+\frac{2}{7})\cdot (1+\frac{2}{9})\cdot (1+\frac{2}{11})\cdot (1+\frac{2}{13})\cdot (1+\frac{2}{15})\cdot (1+\frac{2}{17})\cdot(1+\frac{2}{19})=\\\\=\frac{5}{3}\cdot \frac{7}{5}\cdot \frac{9}{7}\cdot\frac{11}{9}\cdot \frac{13}{11}\cdot\frac{15}{13}\cdot\frac{17}{15}\cdot \frac{19}{17}\cdot \frac{21}{19}=\\\\=\frac{21}{3}=7[/tex]
c)
[tex](1+\frac{2}{3})\cdot(1+\frac{2}{5})\cdot (1+\frac{2}{7})\cdot \ldots\cdot (1+\frac{2}{2017})=\\\\=\frac{5}{3}\cdot \frac{7}{5}\cdot \frac{9}{7}\cdot\ldots\cdot\frac{2019}{2017}=\\\\=\frac{2019}{3}=673[/tex]
d)
[tex](1+\frac{1}{2})\cdot(1+\frac{1}{3})\cdot (1+\frac{1}{4})\cdot\ldots\cdot (1+\frac{1}{2019})=\\\\=\frac{3}{2}\cdot\frac{4}{3}\cdot\frac{5}{4}\cdot\ldots\cdot\frac{2020}{2019}=\\\\=\frac{2020}{2}=1010[/tex]
