# Solving summation problem

Greetings.

$$\sum_{i=x}^{2x} 5 - \sum_{k=2x+1}^{5x+2} 5 = -25$$

How can I discover the value of X?

EDIT: Turns out it was 5x+2, not 5x+1. My bad, apologies for the trouble.

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You asked a similar question, so I think you didn't understand Derek's comment. Here we need to count how many integers there are between $x$ and $2x$ and between $2x+1$ and $5x+2$. Since you are not familiar with these things, try to write all possible numbers. You'll get something like $$x,\ x+1, \ldots, \ x+ (x-1), \ 2x$$ and $$2x+1, \ldots, \ 2x + (x-1), \ 3x, \ 3x+1, \ldots, \ 3x+ (x-1), \ 4x,$$ $$4x+1, \ldots, \ 4x +(x-1), \ 5x, \ 5x+1, \ 5x+2.$$ Now it's easy to see that you have $x+1$ numbers between $x$ and $2x$ and $3x+2$ between $2x+1$ and $5x+2$.

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Use the fact that

$\sum_{i=x}^{2x} 5 = 5 \cdot (2x - x) = 5x$

and

$\sum_{k=2x+1}^{5x+1} 5 = 5 \cdot (5x+1-2x-1) = 15x$

Then, the equation becomes

$5x - 15x = -25$

which is easy to solve, you just get $x = 25/10$ or in simpler terms $x = 5/2$

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I'm sorry, I corrected the problem. And this is what I was looking for, I couldn't remember what to do in this situation. Much appreciated. – Qosmo Oct 31 '10 at 20:42
You might want to update your answer just for the sake of correctness. Thanks again. – Qosmo Oct 31 '10 at 20:51