A linear pattern in Intermediate Algebra is a sequence that changes by the same amount each step. That constant change makes it an arithmetic sequence and can be written with an explicit formula.
A linear pattern in Intermediate Algebra is a number pattern where the change from one term to the next stays the same. If you add the same amount each time, or subtract the same amount each time, the sequence is linear in the sense that its terms grow at a constant rate.
The most common example is an arithmetic sequence. For instance, 4, 7, 10, 13 is a linear pattern because each term increases by 3. That repeated increase is called the common difference. If the pattern goes down, the common difference is negative, like 20, 16, 12, 8, where the sequence decreases by 4 each time.
The reason this counts as a linear pattern is that the relationship between the term number and the value of the term is steady. If you graph the term number on the x-axis and the term value on the y-axis, the points line up in a straight-line pattern. That does not mean the sequence itself is a line, but it does mean the rule behind it is linear.
In Intermediate Algebra, you usually move between three ways of describing the same pattern. You can list the terms, describe the pattern in words, or write a formula. The explicit formula for an arithmetic sequence, a_n = a_1 + (n - 1)d, is the cleanest way to show the pattern because it tells you any term without listing every previous one.
A quick way to check for a linear pattern is to compare consecutive terms. If the differences are the same every time, you have a constant rate of change. If the differences change, the pattern is not linear, even if the numbers look tidy at first glance. For example, 2, 4, 8, 16 is not a linear pattern because each step multiplies by 2 instead of adding the same amount.
Linear pattern is one of the first places Intermediate Algebra connects sequences to functions. Once you can spot a constant difference, you can predict future terms, write formulas, and solve for missing values without guessing.
It also sets up a bigger idea you will use again and again: constant change. In this course, constant change shows up in arithmetic sequences, tables of values, and straight-line graphs. When you see a pattern that adds the same amount each step, you are really looking at a basic linear relationship in sequence form.
This term matters because it gives you a fast way to check whether a pattern fits an arithmetic sequence. That can save a lot of time on homework problems that ask you to continue a sequence, find a specific term, or write a rule from a list of numbers.
It also helps you avoid confusing linear growth with exponential growth. A savings plan that adds $25 each week is linear pattern behavior. A balance that grows by 25% each week is not. That difference shows up all over algebra, and noticing it early makes later units much easier.
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view galleryArithmetic Sequence
A linear pattern in this course is usually an arithmetic sequence. The sequence is arithmetic when each term is found by adding or subtracting the same number. If you can identify that repeated change, you can name the sequence and start building formulas for future terms.
Common Difference
The common difference is the number you add or subtract each time in a linear pattern. It is the quickest way to test whether the pattern is constant. If the difference stays the same between every pair of consecutive terms, the sequence has a linear structure.
Explicit Formula
The explicit formula is how you write a linear pattern so you can find any term directly. Instead of working step by step, you plug in the term number and calculate the value. That makes it especially useful when a problem asks for a far-away term, like the 20th or 50th term.
Recursive Formula
A recursive formula describes a linear pattern by giving the first term and the rule for getting the next term. That is different from the explicit formula, which jumps straight to any term. Recursive rules are useful when you want to show how the pattern grows one step at a time.
A quiz or problem set might give you a list of numbers and ask whether the pattern is linear, then ask you to find the common difference and write a formula. You may also need to fill in a missing term, decide if a table is arithmetic, or explain why a pattern is not linear. The main move is checking consecutive differences. If they stay constant, you can use the arithmetic sequence formula to find terms, and if they do not, you know the pattern is not linear. On word problems, look for repeated addition or subtraction in the situation, such as a weekly savings plan or a steady increase in distance.
A linear pattern adds or subtracts the same amount each step, while an exponential pattern multiplies by the same factor each step. They can look similar at first, especially in tables, but the change test is different: subtraction checks linear patterns, and ratios or multiplication checks exponential patterns.
A linear pattern in Intermediate Algebra changes by the same amount from one term to the next.
Most linear patterns in this course are arithmetic sequences with a constant common difference.
If the differences between consecutive terms are not the same, the pattern is not linear.
You can describe a linear pattern with an explicit formula, a recursive rule, or a list of terms.
A steady add-or-subtract pattern is linear, while repeated multiplication points to something else.
A linear pattern is a sequence where each term changes by the same amount. In Intermediate Algebra, that usually means an arithmetic sequence, like 3, 6, 9, 12. The constant change is what makes it linear.
Subtract each term from the one after it and check whether the differences match. If they do, the sequence has a constant difference and is linear. If the differences change, the sequence is not linear.
In this course, the terms are often used almost the same way. A linear pattern describes the constant change, and an arithmetic sequence is the sequence format that follows that pattern. If the common difference stays the same, you are dealing with both.
You use it to find missing terms, write an explicit formula, or predict later terms. If a problem shows repeated addition or subtraction, the sequence is probably linear. That lets you work with the common difference instead of guessing the next numbers.