Linear Sequences

A linear sequence is a list of numbers that changes by the same fixed amount from one term to the next. Because that change never varies, the terms of a linear sequence always line up on a straight line when you plot them — which is exactly where the name comes from.

For example, look at the list \(2, 5, 8, 11, 14, \dots\) Each term is 3 more than the one before it. That constant jump of 3 makes this a linear sequence. In most algebra courses, this same idea is called an arithmetic sequence, so you can use the terms interchangeably.

The fixed amount added (or subtracted) between consecutive terms is called the common difference, usually written as \(d\). To find it, subtract any term from the one right after it:

\(d = a_{n} - a_{n-1}\)

In the sequence \(2, 5, 8, 11, 14, \dots\), \(d = 5 - 2 = 3\). If \(d\) is positive, the sequence increases; if \(d\) is negative, the sequence decreases, such as \(20, 16, 12, 8, \dots\) where \(d = -4\).

To confirm a sequence is linear, check that the difference between every pair of consecutive terms stays exactly the same. If the difference changes, the sequence is not linear.

A recursive formula defines each term using the term that came right before it. For a linear sequence, this is simply:

\(a_n = a_{n-1} + d\)

You also need to state the first term, \(a_1\), since the recursive rule cannot run without a starting point. For \(2, 5, 8, 11, 14, \dots\), the recursive formula is \(a_1 = 2\), \(a_n = a_{n-1} + 3\).

The explicit formula, often called the nth term formula, lets you calculate any term directly without listing every term before it. It is built from the first term \(a_1\) and the common difference \(d\):

\(a_n = a_1 + (n - 1)d\)

For the sequence \(2, 5, 8, 11, 14, \dots\), \(a_1 = 2\) and \(d = 3\), so the formula becomes:

\(a_n = 2 + (n - 1)(3) = 3n - 1\)

To find the 20th term, substitute \(n = 20\):

\(a_{20} = 3(20) - 1 = 59\)

Notice that the formula \(a_n = 3n - 1\) has the same structure as a linear equation \(y = mx + b\), where the common difference \(d\) plays the role of the slope and \(a_1 - d\) plays the role of the \(y\)-intercept.

If you plot the term number \(n\) on the horizontal axis and the term value \(a_n\) on the vertical axis, every point falls on the same straight line. This visual check is a quick way to confirm a sequence is linear.

Suppose a sequence begins \(7, 4, 1, -2, \dots\)

Step 1: Find the common difference. \(d = 4 - 7 = -3\).

Step 2: Write the explicit formula. \(a_n = 7 + (n - 1)(-3) = -3n + 10\).

Step 3: Use the formula to find the 15th term. \(a_{15} = -3(15) + 10 = -35\).

Once you can write both formulas for a linear sequence, you are also ready to add up its terms; that skill is covered separately in arithmetic series. If a pattern instead grows by multiplying by a constant ratio rather than adding a constant difference, it is not linear at all — it belongs to geometric sequences instead.

Before you accept a sequence as linear, run through this checklist:

1. Subtract consecutive terms to check the difference is the same every time.

2. Identify \(a_1\), the first term, and \(d\), the common difference.

3. Build the explicit formula \(a_n = a_1 + (n-1)d\) to jump to any term you need.

4. Build the recursive formula \(a_n = a_{n-1} + d\) when you only need the next term.

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