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So, it's been a long time since I've studied math, so I'm having more trouble with this problem than I thought I would as for some help. I have an arithmetic sequence $0,...,99$ with the difference being $1$. Basically I have numbers $0$ to $99$. The sum of this sequence is $4950$. If I then say that the sum $1797 = 0,...,n$, how would I find $n$?

I've gotten to the point in my equation where $2(1797) = n * (n+1)$ but I don't know where to go from here.

Also, obviously, there is no whole number solution to this particular issue, however there is a rational one and that's what I am looking for.

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    Perhaps "rational" is not the right word... It has been awhile.2012-11-16

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Use simple formula for quadratic equations. Re-writing your equation you get $n^2 + n - 2*1797 = 0$. The number by $n^2$ is customarily named $a$, the one by $n$ is $b$ and the third one $c$.

There are two solutions given by: $\frac{-b\pm\sqrt{b^2-4ac}}{2a}$ which gives us both solutions i.e. $\frac{-1\pm\sqrt{14377}}{2}$

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    Ok, so rational's not the right term. But both of those examples are the kind of thing I needed.2012-11-16
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You are right you want to solve $2\cdot 1797=n(n+1)$, which is $n^2+n-3594$. You can look to factor this or use the quadratic equation to get $n=\frac{-1\pm\sqrt{1+4\cdot 3594}}2$ Which has solutions about $-60.5$ and $59.5$. Neither of these is rational as $14377$ is not a square.

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    It's been awhile... basically I was looking for the larger one and I g$u$ess rational isn't the right term to use.2012-11-16
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We typically use $n$ to denote (non-negative) integers, but that isn't a huge deal.

If you're talking about an arithmetic sequence with difference $1$ starting at $0$, then every number in the sequence will be a non-negative integer. There's no avoiding that.

If we're not constrained to a difference of $1$, then you may as well just take the sequence to be $0,1797$. There's really no need to to get into sequences with non-integer rational difference.

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    I understand what you're trying to do. Here's the kicker, though: what does "..." mean in the context of $1,...,x$ for arbitrary $x$? There's a natural interpretation when $x$ is a positive integer--namely, that we're talking about an arithmetic sequence with difference $1$. You've already noticed, though, that there's no integer solution to $x^2+x=2(1797)$--in fact, there isn't even a rational solution (not that it would make sense, anyway).2012-11-15