**Please Note: **
This article is written for users of the following Microsoft Excel versions: 2007, 2010, 2013, 2016, 2019, and Excel in Microsoft 365. If you are using an earlier version (Excel 2003 or earlier), *this tip may not work for you*. For a version of this tip written specifically for earlier versions of Excel, click here: Finding the Sum of a Sequential Integer Range.

Written by Allen Wyatt (last updated June 8, 2023)**This tip applies to** Excel 2007, 2010, 2013, 2016, 2019, and Excel in Microsoft 365

Excel includes the FACT worksheet function which returns the factorial of a value. (The factorial of the number X is the result of multiplying 1 * 2 * 3 ... * X.) Sabeesh wonders if there is a similar function that will return the sum of the values (1 + 2 + 3 ... + X) instead of the result of the values.

There is no such function built into Excel, but a quick mathematical formula will do the trick. The proper terminology to refer to this type of sum is a "triangular number." This derives from the fact that if the sum was represented with objects, they could always be arranged in the form of a triangle. For example, if you had 5 objects on the bottom row, 4 on the next, 3 three on the third, 2 on the fourth, and 1 on the top row, you have a triangle. Summing the number of objects (5 + 4 + 3 + 2 + 1) is what Sabeesh wants to do.

The answer to this problem can be expressed as a mathematical formula, reportedly discovered by Carl Friedrich Gauss. (Which is the source for another name of this type of number: a Gaussian Summation.) Note that the sum of opposite rows in the above example are always the same: 5 + 1 is the same as 4 + 2. This is true regardless of the number of rows; if there were 100 rows, then 100 +1 is the same result as 99 + 2, 98 + 3, 97 + 4, etc. What you end up with is 50 "pairs" of numbers equal to 1 more than the upper limit of your range.

The upshot of all this—without going through a lot of explanation—is that you can find the triangular number for any positive value (where you start at 1 and end with X) in the following manner:

=X*(X+1)/2

Thus, if you had a number in cell A1 and you wanted to know the sum of the range of 1 through that number, you could use this formula:

=A1*(A1+1)/2

This formula provides a simple way to determine the sum required, without the necessity of resorting to using a macro.

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This tip (9998) applies to Microsoft Excel 2007, 2010, 2013, 2016, 2019, and Excel in Microsoft 365. You can find a version of this tip for the older menu interface of Excel here: **Finding the Sum of a Sequential Integer Range**.

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2023-06-10 06:26:27

Mike J

This is an old thread, but I think your first formula should be

=(Y*(Y+1) - X*(X-1))/2

assuming one is summing the numbers X to Y inclusive.

2019-06-29 19:20:43

Bill Multack

If A1 and A2 contain the intergers (A2 larger) then the entry to find the sum from A1 to A2 could look like this: =(Y^2 + Y -X^2 + X)/2.

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