Consider a sequence of numbers where the first term is $a_1$, the second term is $a_2$, the third term is $a_3$, and so on. The common difference, or constant that is being added to each term is $d$, and $a_n$ is the $n^{\rm th}$ term.

A pattern (formula) starts to emerge. Using the previous information, we obtain:

$a_1=a_1$
$a_2=a_1+d$
$a_3=a_2+d=a_1+d+d=a_1+2d$
$a_4=a_3+d=a_1+2d+d=a_1+3d$
$a_5=a_4+d=a_1+3d+d=a_1+4d$
$\vdots$
$a_n=a_1+(n−1)d$

This is the formula for the $n^{\rm th}$ term of an arithmetic sequence.

Example 1: Consider the sequence of numbers $−12, ~−9, ~−6, ~−3,~0$ $\ldots$ Find the $\rm 126^{th}$ term. That is to say, find $a_{126}$.

We know $a_1=−12$, $d=3$, and $n=126$ since we are looking for the $\rm 126^{th}$ term. Using the formula for the $n^{\rm th}$ term of an arithmetic sequence,

$a_n=a_1+(n−1)d$

$a_{126}=−12+(126−1)(3)$ $=−12+(125)(3)$ $=−12+375 =363$. Thus the 126th term in this sequence will be $363$.

If we wanted to determine the sum of all $126$ terms, then another pattern (formula) emerges. Without going into a formal proof, the sum of the first $n$ terms, $\mathrm S_n$, of an arithmetic sequence, called an arithmetic series, is given by: $\mathrm S_n=\dfrac{n}{2}(2a_1+(n−1)d)$. Therefore, we need to know a few pieces of information. Namely, the first term $a_1$, how many terms are we summing $n$, and the common difference $d$.

Notice that $2a_1=a_1+a_1$. Thus, $\mathrm S_n$ can also be written as:

$\mathrm S_n=\dfrac{n}{2}(a_1+a_1+(n−1)d)$
$\mathrm S_n=\dfrac{n}{2}(a_1+a_n)$

Therefore, we need to know a few different pieces of information. Namely, the first term $a_1$, how many terms are we summing $n$, and the last term in the sequence $a_n$.

Example 2: Marie is training for a race. The first week of the year she runs $\rm 3~km$. The next week of the year she runs $\rm 3.5~km$. The next week of the year she runs $\rm 4~km$. This progression continues for one year ($52$ weeks). Determine the total number of km Marie ran during the year.

Since we are being asked to sum an arithmetic sequence, or calculate an arithmetic series, we can use the formula:

$\mathrm S_n=\dfrac{n}{2}(2a_1+(n−1)d)$

It is clear that $a_1=3$ and $d=0.5$ and the number of terms is $n=52$. This gives us a result of:

$\mathrm S_{52}=\dfrac{5}{22}(2(3)+(52−1)(0.5))$
$\mathrm S_{52}=26(6+51(0.5))$
$\mathrm S_{52}=26(6+25.5)$
$\mathrm S_{52} =26(31.5)$
$\mathrm S_{52}\rm =819~km$.