Question

(1pe 1×2 + 1 pe 2×3 +...+ 1 pe 19 ×20 ) : [(19 pe 4 ) la puterea 5 : (19pe4) la puterea 2 : (19pe4)la puterea 2 ] = ????

Answer 1

[tex]( \frac{1}{1*2} + \frac{1}{2*3} +...+ \frac{1}{19*20} ):[ (\frac{19}{4} )^{5} : (\frac{19}{4} )^{2}: (\frac{19}{4} )^{2}]=\\\\( \frac{2-1}{2*1} + \frac{3-2}{3*2}+...+ \frac{20-19}{20*19}) :(\frac{19}{4} )^{5-2-2}=\\\\(1- \frac{1}{2} + \frac{1}{2} - \frac{1}{3} +....+ \frac{1}{19} - \frac{1}{20} ): \frac{19}{4} =\\\\(1- \frac{1}{20} ):\frac{19}{4}=\\\\ \frac{19}{20}: \frac{19}{4} =\frac{19}{20}*\frac{4}{19}= \frac{1}{5} \\\\\\\\ \frac{b-a}{a*b} = \frac{1}{a} - \frac{1}{b}[/tex]