A series is the sum of the terms of a sequence. $S_n$ refers to the sum of the first $n$ terms of a sequence.

The Sum of an Arithmetic Sequence

There are two formulas that are commonly used to calculate the sum of an arithmetic sequence:

$S_n=n\Big(\frac{u_1+u_n}{2}\Big)$

where $n$ is the number of terms; $u_1$ is the first term and $u_n$ is the last term in the sequence.

$S_n=\frac{n}{2}\Big(2u_1+(n-1)d\Big)$

where $n$ is the number of terms; $u_1$ is the first term and $d$ is the common difference between terms.

Example sum

That is, the formula $S_n=\dfrac{n(u_1+u_n)}{2}$ is most useful when you have the first term, the last term, and the number of terms.

The formula $S_n=\dfrac{n}{2}(2u_1+(n-1)d)$ is most useful when you have the first term, the common difference, and the number of terms.

Calculate:

The sum of the first 12 positive multiples of 5. (That is, 5, 10, 15 … 55, 60).
The sum of the first 20 terms of the arithmetic sequence beginning 9, 16, 23 …
The sum of the first 41 terms of the arithmetic sequence beginning 80, 76, 72 …

Show Answers

The first formula is similar to the formula to find the area of a trapezoid.

https://www.mathsisfun.com/geometry/trapezoid.html#Area

The trapezoid-type shape below is made from the terms of an arithmetic sequence. Move the slider to see how the trapezoid-type shape rearranges to a rectangle.

Applet link

Notice that that the base of the rectangle is calculated as the average of the first and the last term. The height is the number of terms in the sequence.

Fit two trapezoids together, then divide by 2

Another way to consider this sum is to create a second trapezoid-like shape with the terms of the sequence. The second shape is congruent to the first and fits like a puzzle to form a larger rectangle:

Applet link

Now we create an area that is double the required sum. We see that the height of the rectangle is the number of terms in the sequence, in this case 7, and the base of the rectangle is the first term plus the last term $4+22=26$ .

We calculate that double the sum of the first seven terms is:

$2S_7=(4+22)\times 7 = 182$

Divide both sides by 2 to calculate the sum required:

$S_7=\dfrac{4+22}{2}\times 7=91$

Historical note

A mathematician named Karl Frederich Gauss has been called ‘the prince of mathematics’ – due to his many contributions to mathematics. While still a primary school child, he impressed his teachers by calculating that $1 + 2+3+ \dots +99 + 100=5050$ . His teacher had given the class the task, expecting it to take some time. But with Gauss in the room, the task was over in seconds! The website www.nrich.maths.org has a nicely written article on this story.