Question

rezolvați :
$\frac{x + 1}{2} + \frac{x + 2}{3} + \frac{ x+ 3}{4} + ... + \frac{x + 2012}{2013} = 2012$
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Answer 1

$\sum\limits_{k=1}^{2012} \dfrac{x+k}{k+1} = 2012 \\ \\ \sum\limits_{k=1}^{2012} \dfrac{(x-1)+(k+1)}{k+1} = 2012 \\ \\ \sum\limits_{k=1}^{2012} \Big(\dfrac{x-1}{k+1} +1 \Big) = 2012 \\ \\ \sum\limits_{k=1}^{2012} \Big(\dfrac{x-1}{k+1} \Big)+ 2012 = 2012 \\ \\ (x-1)\cdot \sum\limits_{k=1}^{2012} \dfrac{1}{k+1} = 2012-2012 \\ \\ (x-1)\cdot \sum\limits_{k=1}^{2012} \dfrac{1}{k+1} = 0 \\ \\ x-1 = \dfrac{0}{\sum\limits_{k=1}^{2012} \dfrac{1}{k+1}} \\ \\ x-1 = 0 \Rightarrow \boxed{x = 1}$