|
|
|
|
|
|
|
||
|
|
|
|
|
Proof of Arithmetic Summation Formula On an intuitive level, the formula for the sum of a finite arithmetic series says that the sum of the entire series is the average of the first and last values, times the number of values being added.
This makes sense, especially if you think of a summation visually as being the sum of the areas of the pictured bars. Copyright © Elizabeth Stapel 2006-2008 All Rights Reserved
Since the bars grow by a fixed amount at each step, you can, in effect, "average" the bars to get the total area:
To prove this equality properly, however, requires a bit more work. We proceed by induction: Proof: Let n = 2. Then we have:
For n = k, assume the following:
Let n = k + 1. Then we have:
By nature of arithmetic sequences, we have: ak = ak+1
– d Then, substituting the above into the n = k + 1 expression, we have:
Therefore the result holds for n = k + 1, and the formula is proved for all n > 2.
|
|
|
|
Copyright © 2006-2008 Elizabeth Stapel | About | Terms of Use |
|
|
|
|
|
|
![(n/2)(a-sub-1 + a-sub-n) = n [ (a-sub-1 + a-sub-n) / 2 ]](series/sum35.gif){kind=small}
![= (ka_1 + ka_(k+1) + a_1 + a_(k+1)) / 2 = [ (k + 1) / 2 ] [a_1 + a_(k+1)]](series/sum39.gif)
