Derivation of the Arithmetic Series Formula
In this lesson, we are going to derive the Arithmetic Series Formula. This is a good way to appreciate why the formula works.
Suppose we have the following terms where [latex]\large{d}[/latex] is the common difference.
first term = [latex]\large{a}[/latex]
second term = [latex]\large{a+d}[/latex]
third term = [latex]\large{a+2d}[/latex]
…
nth term or last term = [latex]\large{a + \left( {n – 1} \right)d}[/latex]
If we add the terms together, we get

Let’s add the terms in reverse or descending order.

Here’s the “trick”. Now, we sum up the two arithmetic series above – the ones with ascending and descending terms. Notice that the sum of each column is always [latex]{2a+(n-1)d}[/latex].

Notice that there are [latex]\large{n}[/latex] numbers of [latex]{2a+(n-1)d}[/latex] on the right side of the equation. Therefore, we can rewrite the equation above as

Finally, solve [latex]\large{S_n}[/latex] by dividing both sides of the equation by [latex]2[/latex].
![Sn=n/2[2a+(n-1)d]](https://www.chilimath.com/wp-content/uploads/2021/03/alternative-arithmetic-series-formula-.png)
The formula above is the expanded version of the arithmetic series formula. We can manipulate it to make it a bit more compact.
Let’s rewrite [latex]2a[/latex] as [latex]\color{blue}a+a[/latex].
![Sn=n/2[a+a+(n-1)d]](https://www.chilimath.com/wp-content/uploads/2021/03/write-2a-as-a-plus-a.png)
Notice that [latex]a+(n-1)d[/latex] is actually the Arithmetic Sequence Formula.
![Sn=(n/2)[a+a+(n-1)d]](https://www.chilimath.com/wp-content/uploads/2021/03/nth-term-formula-within.png)

Simplify the formula by substituting the expression [latex]a+(n-1)d[/latex] by [latex]\large{a_n}[/latex]. This is the standard form of the Arithmetic Series Formula.

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