# Shortcut method to sum n consecutive numbers [duplicate]

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.): $$10 + 11 + 12 + 13 = 46$$.

• 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...... Commented Jul 9, 2011 at 9:45
• Alternately, if there are $n$ successive integers, starting with $m+1$, the answer is $mn + (1 + 2 + \ldots + n)$. Commented Jul 9, 2011 at 10:02
• I'm pretty sure this is a duplicate. :) Commented Jul 9, 2011 at 10:15
• BTW: MarkUp is not allowed in subject title of posts. I've removed the asterisks. Commented Jul 9, 2011 at 11:15
• 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.
– B T
Commented Dec 18, 2015 at 8:10