Summation or Algebra? I am trying to visualize the concept that should be involved with this problem: What is $x$ in the equation 
$$(x-1)-2(x-2)+3(x-3)-4(x-4)+\dots-10(x-10)=0\quad ?$$
What should I do? Express in summation (which I find very hard) or just algebra?

$$(x-1)-2(x-2)+3(x-3)-4(x-4)+\dots-10(x-10)=0$$
$$\implies x(1-2+3-4+\cdots-10)-(1-2^2+3^2-\cdots-10^2)=0$$
$$\implies x=\dfrac{1-2^2+3^2-\cdots-10^2}{1-2+3-4+\cdots-10}$$
$$\implies x=11$$

$$1-2^2+3^2-\cdots-10^2=\sum_{n=1}^{10}(-1)^{n-1}n^2=-55$$
and $$1-2+3-4+\cdots-10=\sum_{n=1}^{10}(-1)^{n-1}n=-5$$
