Question

fie.nr a =(1-1/2)×(1-1/3)...×(1-1/9) si b= (1+1/2)×(1+1/3) ×...×(1+1/8) atunci 2ab este

Answer 1

$\displaystyle\\ a=\Big(1-\frac{1}{2}\Big)\Big(1-\frac{1}{3}\Big)\Big(1-\frac{1}{4}\Big)\Big(1-\frac{1}{5}\Big)\Big(1-\frac{1}{6}\Big)\Big(1-\frac{1}{7}\Big)\Big(1-\frac{1}{8}\Big)\Big(1-\frac{1}{9}\Big)=\\\\\\ a=\frac{1}{2}\times \frac{2}{3}\times \frac{3}{4}\times \frac{4}{5}\times \frac{5}{6}\times \frac{6}{7}\times \frac{7}{8}\times \frac{8}{9}=~~~~~\text{Simplificam.}\\\\\\ \boxed{\bf a= \frac{1}{9}}$

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$\displaystyle\\ b=\Big(1+\frac{1}{2}\Big)\Big(1+\frac{1}{3}\Big)\Big(1+\frac{1}{4}\Big)\Big(1+\frac{1}{5}\Big)\Big(1+\frac{1}{6}\Big)\Big(1+\frac{1}{7}\Big)\Big(1+\frac{1}{8}\Big)=\\\\ b=\frac{3}{2}\times\frac{4}{3}\times\frac{5}{4}\times\frac{6}{5}\times\frac{7}{6} \times\frac{8}{7}\times\frac{9}{8}=~~~\text{Simplificam.}\\\\ b=\frac{1}{2}\times\frac{9}{1}=\frac{9}{2}\\\\ \boxed{\bf b=\frac{9}{2}}\\\\\\ 2ab=2\times\frac{1}{9}\times\frac{9}{2}=\frac{2\times1\times9}{9\times2}=\frac{18}{18}=\boxed{\bf 1}$