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Proof of the Arithmetic Summation Formula

Purplemath

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:

n = 1, 2, 3,... and a_i = a_1, a_2, a_3,...

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

animation: short bars grow, tall bars shrink, meeting in the middle at the average value

(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₊₁ – d

a_k₊₁ = a1 + kd

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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URL: https://www.purplemath.com/modules/series6.htm

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Proof of the Arithmetic Summation Formula

Purplemath

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