Question

1+1/2+1/3+1/4+...+1/2017+1/2+2/3+3/4+...+2016/2017 rezolvarea vs rog

Answer 1

   
[tex]\displaystyle\\ 1+ \left(\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\cdots+\frac{1}{2017}\right)+\left(\frac{1}{2}+\frac{2}{3}+\frac{3}{4}+\cdots+\frac{2016}{2017}\right) \\ \\ \\ \text{Fiecare paranteza are cate 2016 termeni.} \\ \text{Mai avem un termen inainte de prima paranteza. Acesta este = 1.} \\ \\ \texttt{Rezolvare:}\\\\ \text{Termenii din paranteza ii grupam cate doi, unu din prima paranteza}\\ \text{si celalalt din a doua paranteza.} [/tex]



[tex]\displaystyle \\ 1+ \left(\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\cdots+\frac{1}{2017}\right)+\left(\frac{1}{2}+\frac{2}{3}+\frac{3}{4}+\cdots+\frac{2016}{2017}\right)= \\ \\ \\ =1+\left(\frac{1}{2}+\frac{1}{2}\right) + \left(\frac{1}{3}+\frac{2}{3}\right) +\left(\frac{1}{4}+\frac{3}{4}\right) +\cdots +\left(\frac{1}{2017}+\frac{2016}{2017}\right)= [/tex]


[tex]\displaystyle \\ =\underbrace{1+\frac{2}{2}+\frac{3}{3}+\frac{4}{4}+\cdots + \frac{2017}{2017}}_{\texttt{In total 2017 termeni}}= \underbrace{1+1+1+1+\cdots +1}_{\texttt{In total 2017 termeni}} = \boxed{2017}[/tex]