Summation or Algebra?

Question

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?

Answer 1

Note that by using Gauss' trick $$\begin{align}(x-1)-10(x-10)&=-9(x-11)\\ -2(x-2)+9(x-9)&=7(x-11)\\ 3(x-3)-8(x-8)&=-5(x-11)\\ -4(x-4)+7(x-7)&=3(x-11)\\ 5(x-5)-6(x-6)&=-(x-11) \end{align}$$ and each term is zero for $x=11$.

Answer 2

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

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

Answer 3

One thing you can do with problems like this is look for helpful patterns. There is clearly a pattern involved, but does it simplify into something you can work with. The first few terms come out as

$$x-1=0$$$$-x+3=0$$$$2x-6=0$$$$-2x+10=0$$$$3x-15=0$$$$-3x+21=0$$Does that help you to see what is going on.