Arithmetic Sequences

Your school cafeteria charges $2.50 per day for lunch. On day 1 you have spent $2.50. On day 2, $5.00. On day 3, $7.50. Each day, the total climbs by exactly $2.50 — the same amount, no exceptions. That fixed, repeating gap is the heart of an arithmetic sequence.

A sequence is a list of numbers in a specific order. Each number in the list is called a term. The first term is $a_1$, the second is $a_2$, and so on. In an arithmetic sequence, you get from one term to the next by adding the same fixed number every single time. That fixed number is called the common difference, and we label it $d$.

In the lunch example, the sequence is $2.50, 5.00, 7.50, 10.00, \ldots$ and the common difference is $d = 2.50$. To find $d$, subtract any term from the term that follows it: $5.00 - 2.50 = 2.50$. That works for any consecutive pair.

The common difference can be negative. If you start with $40 in a gift card and spend $8 each visit, the balances form the sequence $40, 32, 24, 16, \ldots$ with $d = -8$. The sequence is decreasing, but the gap is still constant.

Once you know the first term and the common difference, you can find any term without listing every term in between. Think about where each term comes from. The first term is just $a_1$. The second term is $a_1 + d$. The third is $a_1 + 2d$. The fourth is $a_1 + 3d$. The pattern: the $n$th term has $d$ added exactly $n - 1$ times. That gives the explicit formula:

This formula lets you jump straight to the 50th term or the 200th term without computing every term before it. Plug in $n$, $a_1$, and $d$, and you are done.

Here is a straightforward example. The sequence $3, 7, 11, 15, \ldots$ has a first term of $a_1 = 3$ and a common difference of $d = 4$. Find the 20th term.

Now a harder case. You know two terms of an arithmetic sequence but not the first term or the common difference. The 4th term is 17 and the 9th term is 37. Find the explicit formula.

Look at that final formula: $a_n = 4n + 1$. Compare it to the slope-intercept form of a line: $y = mx + b$. The common difference $d = 4$ plays the same role as slope — it is the constant rate of change. The expression $a_1 + (n-1)d$, when distributed and simplified, always produces a linear expression in $n$. Arithmetic sequences and linear functions are the same mathematical structure. A sequence is just a linear function whose input is restricted to positive whole numbers.

The graph below plots the terms of $a_n = 4n + 1$ as discrete points, alongside the line $y = 4x + 1$. The points sit exactly on the line, but the sequence only lives at integer values of $n$.