Sum of n consecutive numbers [duplicate]

elementary-number-theory

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


Solution 1:

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

Solution 2:

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:

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