# The Formula for the Sum of Consecutive Integers From 1 to *x*

## To know the sum of consecutive integers, __starting from the number 1,__ you must use & remember this formula:

(x^2 + x)/2 = y

x = the final integer, y = the sum

__Examples:__

1 + 2 + 3 + 4 + 5 + 6 + 7 = 28

(7^2 + 7)/2 = (49 + 7)/2 = 56/2 = 28

In this case, x = 7 & y = 28

1 + 2 + 3 + ... + 100 = 5,050

(100^2 + 10)/2 = (10,000 + 100)/2 = 10,100/2 = 5,050

In this case, x = 100 & y = 5,050

If you have a graphing calculator, then you can insert the function at the top of this Web page into your calculator to see the sums in one column(*y*), and the final integer(s) you chose in the other column(*x*)! You could also see that it's a **quadratic** function. Also, notice that the differences of the sums are consecutive integers!

1 + 2 + 3 + ... + 250 = 31,375

(250^2 + 250)/2 = (62,500 + 250)/2 = 62,750/2 = 31,375

In this case, x = 250 & y = 31,375

## Here's the final & perhaps most interesting example!

1 + 2 + 3 + ... + 666 = 222,111

(666^2 + 666)/2 = (443,556 + 666)/2 = 444,222/2 = 222,111

In this case, x = 666 & y = 222,111

To add more info to Goldilocks' comment, if *x* = 0, then *y* = 0, according to the function. (Zero is an integer, too!) Furthermore, zero & one are the only 2 *idempotent* numbers; they're the only 2 numbers that are their own squares.

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© Derek Cumberbatch