Hi there boschedog-ga,
What you are looking for is the formula for an arithmetic series (an
arithmetic series is the sum of an arithmetic sequence). We call a
sequence of numbers an arithmetic sequence if the difference of any
two consecutive term is the same. An example is a sequence of odd or
even numbers, in which case the common difference is 2, or the
sequence of natural numbers which you listed in your question, in
which case the difference is 1.
Now let's get right to the point.
Assuming that you have a number, eg 8, in cell A1, the formula for the
sum would be: =A1*(A1+1)/2
There are a few ways to understand why this is so.
First method:
Assume that the number you have is "n".
So, you're looking for 1 + 2 + 3 +...+ (n-2) + (n-1) + n. Let's call
this sum, "S".
Write out the sequence again below it, but in reverse order and add them.
S = 1 + 2 + 3 + ... + (n-2) + (n-1) + n
S = n + (n-1) + (n-2) + ... + 3 + 2 + 1
--------------------------------------------------------------
S+S = (n+1) + (n+1) + (n+1) + ... + (n+1) + (n+1) + (n+1)
Now, we can see that the left side is equal to 2S since we just added
the same series to itself. On the right side, you see that every pair
of terms add up to (n+1). We also know that there are n pairs of
(n+1) since S has n terms.
Thus 2S = (n+1)*n
S = n(n+1)/2
Second method:
If you want to see a graphical interpretation of the sum, please check
out this link:
http://merganser.math.gvsu.edu/calculus/summation/arithmetic.html
I hope this has helped and please feel free to ask for any
clarification and I'd be happy to help you further. The formula for
different arithmetic series (where the difference between the terms
isn't 1) requires slight adjustment but can be derived in the same way
- please let me know if you require the general formula.
Cheers,
Tox-ga |