Question

x+ 2x+3x+....+2015x=1+2+3+...+2014

Answer 1

$x+2x+3x+......+2015x=1+2+3+......+2014\\x*(1+2+3+.....+2015)=\frac{2014*(2014+1)}{2}\\x*(\frac{2015*(2015+1)}{2})=\frac{2014*2015}{2}\\x*(\frac{2015*2016}{2})=\frac{4058210}{2}\\x*( \frac{4062240}{2})=2029105\\x*2031120=2029105\\x=\frac{2029105}{2031120}$

Answer 2

$\displaystyle x+2x+3x+...+2015x=1+2+3+...+2014 \\ \\ x(1+2+3+...+2015)= \frac{2014(2014+1)}{2} \\ \\ x \cdot \frac{2015(2015+1)}{2} = \frac{2014 \cdot 2015}{2} \\ \\ x \cdot \frac{2015 \cdot 2016}{2} = \frac{4058210}{2} \\ \\ x \cdot \frac{4062240}{2} =2029105 \\ \\ x \cdot 2031120=2029105 \\ \\ x= \frac{2029105}{2031120}$