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.
ExcelTips is your source for cost-effective Microsoft Excel training. 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
@Bill Multack
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
To find the sum of numbers from one integer to another, one can use this formula: [assumes Y>X] =(Y*(Y-1) - X*(X-1))/2.
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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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:
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