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Possible Duplicate:
Proof for formula for sum of sequence $1+2+3+\ldots+n$?

Is there a shortcut method to working out the sum of n consecutive positive integers?

Firstly, starting at $1 ... 1 + 2 + 3 + 4 + 5 = 15.$

Secondly, starting at any other positive integer ...($10$ e.g.): $10 + 11 + 12 + 13 = 46$.

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    The solution to the second problem follows from a solution of the first since $(m+1)+\cdots+n=S(n)-S(m)$ where $S(k)=1+\cdots+k$. About the first... the legend says that Gauss as a schoolboy realized quickly that if you sum as $(1+n)+(2+(n-1))+\ldots$ all the summands are the same, thus......2011-07-09
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    Alternately, if there are $n$ successive integers, starting with $m+1$, the answer is $mn + (1 + 2 + \ldots + n)$.2011-07-09
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    I'm pretty sure this is a duplicate. :)2011-07-09
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    BTW: MarkUp is not allowed in subject title of posts. I've removed the asterisks.2011-07-09
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    So I thought this was certainly a duplicate. But then I attempted to find such a duplicate, but to no avail. I'm at a loss as to how to effectively search for such a duplicate now.2011-07-09
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    @mixedmath, searching this website is more of an art than a science. Typing "consecutive" into the search box brought up question 15225, which covers some of the same territory but isn't exactly a duplicate.2011-07-10
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    This is not at all a duplicate of the question its marked as a dupe of. This is asking for a simple formula, the other a detailed proof. Very different things. StackExchange has an epidemic of busy bodies marking things as duplicates unnecessarily and destructively.2015-12-18

2 Answers 2

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Take the average of the first number and the last number, and multiply by the number of numbers.

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    ... and this works for any arithmetic progression2011-07-09
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    computation Sum = total_numbers * ( first / 2 + last / 2 )2017-03-14
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The rule, as given by Gerry's answer (and the generalization as per Henry's comment) can be easily visualized, in a similar way as we deduce the area of a rectangular trapezium:

enter image description here

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    Thanks Leon. Great illustration of the math.2011-07-13