Please Note: This article is written for users of the following Microsoft Excel versions: 2007, 2010, and 2013. 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.

Finding the Sum of a Sequential Integer Range

by Allen Wyatt
(last updated July 26, 2014)

4

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, and 2013. You can find a version of this tip for the older menu interface of Excel here: Finding the Sum of a Sequential Integer Range.

Author Bio

Allen Wyatt

With more than 50 non-fiction books and numerous magazine articles to his credit, Allen Wyatt is an internationally recognized author. He  is president of Sharon Parq Associates, a computer and publishing services company. ...

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Comments for this tip:

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What is nine minus 3?

2017-02-22 05:16:05

naseemkanwal

how to -2,-2,-2=-6 & +2,+2,+2=6


2016-04-27 08:55:39

Michael (Micky) Avidan

@Brandon,
Your second formula can be "sinplified" to:
=(A1*(A1+1)-(B1-1)*(B1))/2
--------------------------
Michael (Micky) Avidan
“Microsoft® Answers" - Wiki author & Forums Moderator
“Microsoft®” MVP – Excel (2009-2016)
ISRAEL


2016-04-26 18:28:44

Brandon

Oh I figured it out. A1 contains the user limit, B1 contains the lower limit:

=(A1*(A1+1)/2)-((B1-1)*((B1-1)+1)/2)

Which should be simplified to:

=(A1*(A1+1)/2)-((B1-1)*(B1)/2)

Thank you Allen, helpful article! Sorry for double comment, didn't realize the first comment was on the page for the older versions of Excel, though I guess they're the same formulae for both.


2016-04-26 18:19:51

Brandon

Is there a way to do this between two digits? For instance, the sum of every integer between-and-equal-to 8 and 26 is 323. Is there a way to dynamically refer to two cells with those two or any other integers?


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