The sum, S_{n}, of the first *n* terms of an arithmetic series is given by:

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.

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This makes sense, especially if you think of a summation visually as being the sum of the areas of the bars pictured below:

Since the bars grow by a fixed amount at each step, you can, in effect, "average" the bars to get the total area:

*(The above graphic is animated on the "live" page.)*

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While the pictures are helpful in providing a sense of what is going on, they don't prove anything in the mathematical sense. To prove this formula properly requires a bit more work. We will proceed by induction:

Prove that the formula for the *n*-th partial sum of an arithmetic series is valid for all values of *n* ≥ 2.

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:

*a _{k}* =

*a _{k}*

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. Q.E.D.

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