Question

x(1/2+1/3+1/4+.....+1/10)= 9-1/2-2/3-3/4-...-9/10

Answer 1

$x\cdot\Big(\dfrac{1}{2}+\dfrac{1}{3}+\dfrac{1}{4}+...+\dfrac{1}{10}\Big)= 9-\dfrac{1}{2}-\dfrac{2}{3}-\dfrac{3}{4}-...-\dfrac{9}{10} =\\ \\= 9 - \dfrac{2-1}{2} -\dfrac{3-1}{3} - \dfrac{4-1}{4}-...-\dfrac{10-1}{10} \\ \\ = 9 -\dfrac{2}{2}-\Big(-\dfrac{1}{2}\Big)-\dfrac{3}{3}-\Big(-\dfrac{1}{3}\Big)-\dfrac{4}{4}-\Big(-\dfrac{1}{4}\Big)-...-\dfrac{10}{10}-\Big(-\dfrac{1}{10}\Big) \\ \\ =9-1+\dfrac{1}{2}-1+\dfrac{1}{3}-1+\dfrac{1}{4}-...-1+\dfrac{1}{10} =$

$=9-1-1-\underset{\text{de~9~ori}}{\underbrace{...}}-1 +\dfrac{1}{2}+\dfrac{1}{3}+\dfrac{1}{4}+...+\dfrac{1}{10}= \\ \\ = 9-9+\dfrac{1}{2}+\dfrac{1}{3}+\dfrac{1}{4}+.....+\dfrac{1}{10} = \\ \\ = \dfrac{1}{2}+\dfrac{1}{3}+\dfrac{1}{4}+...+\dfrac{1}{10}\\ \\ \\ \Rightarrow x\cdot\Big(\dfrac{1}{2}+\dfrac{1}{3}+\dfrac{1}{4}+...+\dfrac{1}{10}\Big) = \dfrac{1}{2}+\dfrac{1}{3}+\dfrac{1}{4}+...+\dfrac{1}{10} \\ \\ \Rightarrow \boxed{x=1}$