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 \large{d} is the common difference.

first term = \large{a}

second term = \large{a+d}

third term = \large{a+2d}

nth term or last term = \large{a + \left( {n - 1} \right)d}

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 {2a+(n-1)d}.


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


Finally, solve \large{S_n} by dividing both sides of the equation by 2.


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 2a as \color{blue}a+a.


Notice that a+(n-1)d is actually the Arithmetic Sequence Formula.


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


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