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$$\sum_{i=1}^{x} x + \sum_{i=1}^{x} 2 = 8$$

Trying to discover $x$ here, so I made: $$x(x-1) + 2(x-1) = 8$$

Which should give me the $x$, but instead of an integer I get: $$x = \frac{-1\pm\sqrt{41}}{2}$$

When the answer should be $x=2$, according to the solution.

What am I doing wrong?

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The index $i$ is an integer from $1$ to $\left\lfloor x\right\rfloor $ – Américo Tavares Oct 31 '10 at 21:41
Presumably you replace $\sum_{i=1}^{x} x + \sum_{i=1}^{x} 2$ with $x(x-1) + 2(x-1)$ because you think each part is equal - but it is easy to see that $\sum_{i=1}^{x} x$ is not equal to $x(x-1)$ just by trying a values of $x$. You can always check your steps numerically to try and find where exactly the discrepancy creeps in. – anon Oct 31 '10 at 21:48
Thank you everyone for the insight, I'm trying to get better at math :)) It's all right now. – Qosmo Oct 31 '10 at 21:50
up vote 2 down vote accepted

You have $x$ integers between $1$ and $x$. Look: $1, 2, 3, \ldots, x-1, x$. So you are just not counting properly.

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