A linear sequence is a list of numbers where the gap between each consecutive term is always the same. That constant gap is called the common difference. For example, in the sequence 3, 7, 11, 15, 19, each number is exactly 4 more than the one before it, making the common difference 4. You’ll also see linear sequences called “arithmetic sequences” or “arithmetic progressions,” which are just different names for the same thing.
How a Linear Sequence Works
The defining feature is simple: pick any two consecutive terms, subtract the earlier one from the later one, and you’ll always get the same number. That number is the common difference, usually written as d. If d is positive, the sequence increases. If d is negative, the sequence decreases. If d is zero, every term is identical.
Take the sequence 10, 7, 4, 1, −2. Subtract any term from the one that follows it and you get −3 every time. The common difference is −3, so this is a linear sequence that counts downward.
Finding the nth Term
Once you know a linear sequence’s pattern, you can jump straight to any term without listing every number before it. The nth term formula looks like this:
Term = dn + c
Here, n is the position in the sequence (1st, 2nd, 3rd, etc.), d is the common difference, and c is a constant you calculate from the first term. This works exactly like the slope-intercept form of a straight line (y = mx + b), which is why the sequence is called “linear.”
For example, the sequence 7, 10, 13, 16, 19 has a common difference of 3. Its nth term rule is 3n + 4. To check: plug in n = 1 and you get 3(1) + 4 = 7, which matches the first term. Plug in n = 2 and you get 3(2) + 4 = 10. It works for every position. If you wanted the 100th term, you’d substitute 100 for n: 3(100) + 4 = 304.
How to Identify a Linear Sequence
The quickest test is to calculate the “first differences.” Subtract each term from the next one in the list. If every difference is the same, the sequence is linear. If the differences vary, it’s something else, like a quadratic or geometric sequence.
Say you’re given the sequence 5, 12, 19, 26. The differences are 12 − 5 = 7, 19 − 12 = 7, and 26 − 19 = 7. All identical, so this is linear with a common difference of 7. Now consider 2, 4, 8, 16. The differences are 2, 4, 8, which aren’t constant. That’s not a linear sequence (it’s actually geometric, where each term is multiplied by the same number rather than increased by the same number).
Linear vs. Geometric Sequences
Linear sequences grow by addition. You add the same fixed value to get from one term to the next, which produces steady, uniform growth. Geometric sequences grow by multiplication. You multiply each term by the same fixed value, called the common ratio, which produces exponential growth or decay.
The practical difference is dramatic. A linear sequence like 2, 4, 6, 8, 10 increases by 2 each time and reaches 200 by the 100th term. A geometric sequence like 2, 4, 8, 16, 32, where each term doubles, reaches over 1,000,000,000,000,000,000,000,000,000,000 by the 100th term. Linear sequences stay predictable and manageable. Geometric sequences accelerate fast.
Adding Up the Terms
If you need the sum of the first n terms of a linear sequence, there’s a shortcut that avoids adding every term individually:
Sum = (n / 2) × (first term + last term)
Here, n is how many terms you’re adding. You’re essentially multiplying the number of terms by the average of the first and last terms. For the sequence 3, 7, 11, 15, 19, the sum of all five terms is (5 / 2) × (3 + 19) = 2.5 × 22 = 55. You can verify by adding them directly: 3 + 7 + 11 + 15 + 19 = 55.
If you don’t know the last term, you can use an expanded version of the formula that relies on the common difference instead: Sum = (n / 2) × (2a + (n − 1)d), where a is the first term and d is the common difference.
What the Graph Looks Like
When you plot a linear sequence on a graph, with position numbers along the horizontal axis and term values along the vertical axis, the points form a straight line. This is directly related to why it’s called “linear.” The common difference acts like the slope of the line: a larger common difference makes a steeper line, while a negative common difference makes a line that slopes downward.
One detail worth noting: unlike a standard linear equation, which draws a continuous solid line, a linear sequence produces only individual dots at whole-number positions (1st term, 2nd term, 3rd term, and so on). There’s no value between the 4th and 5th terms, because sequences only exist at whole-number positions. The dots fall along a straight line, but the line itself isn’t filled in. Mathematicians describe this by saying a sequence is “discrete” while a linear function is “continuous.”
Common Examples in Everyday Life
Linear sequences show up whenever something increases or decreases by a fixed amount. Saving $50 every week creates a linear sequence of your total savings: 50, 100, 150, 200, and so on. A building where each floor is 3 meters tall has floor heights of 3, 6, 9, 12 meters. Seat numbers in a row, pages in chapters of equal length, and countdown timers all follow linear patterns.
Recognizing these patterns lets you predict future values without counting every step. If you know the pattern and the starting point, you can calculate any term directly using the nth term formula, or find the total of all terms using the sum formula.