Finding the Sum of a Sequential Integer Range

Please Note: This article is written for users of the following Microsoft Excel versions: 97, 2000, 2002, and 2003. If you are using a later version (Excel 2007 or later), this tip may not work for you. For a version of this tip written specifically for later versions of Excel, click here: Finding the Sum of a Sequential Integer Range.

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 (9997) applies to Microsoft Excel 97, 2000, 2002, and 2003. You can find a version of this tip for the ribbon interface of Excel (Excel 2007 and later) here: Finding the Sum of a Sequential Integer Range.

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Michael (Micky) Avidan    17 Oct 2016, 06:46
@Les McGarry,
Try:
=SUMPRODUCT((MOD(A1:A40,4)=1)*A1:A40)
--------------------------
Michael (Micky) Avidan
“Microsoft® Answers" - Wiki author & Forums Moderator
“Microsoft®” MVP – Excel (2009-2017)
ISRAEL
Les McGarry    17 Oct 2016, 01:58
If I have a column (or row) of numbers from A1 to A40 and want to sum the first and every 4th number (ie. A1, A5, A9, etc) is there a formula to do this?
Michael (Micky) Avidan    27 Apr 2016, 05:49
@Brandon,
Try my suggested 'Array Formula' in the following picture:
(see Figure 1 below)
--------------------------
Michael (Micky) Avidan
“Microsoft® Answers" - Wiki author & Forums Moderator
“Microsoft®” MVP – Excel (2009-2016)
ISRAEL

Figure 1.

Brandon    26 Apr 2016, 18:17
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?
Rex    06 Jan 2016, 17:51
Thank you!

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