Common difference is the constant amount added or subtracted between consecutive terms in an arithmetic sequence. In College Algebra, it is usually written as d.
Common difference is the fixed change from one term to the next in an arithmetic sequence. In College Algebra, that means every step in the pattern is the same size, so you can predict later terms without listing them one by one.
If the sequence goes up by the same amount each time, the common difference is positive. If it goes down by the same amount each time, the common difference is negative. For example, in 4, 7, 10, 13, ... the common difference is 3 because each term is 3 more than the one before it. In 20, 15, 10, 5, ... the common difference is -5 because each term drops by 5.
You can find the common difference by subtracting a term from the term that comes after it: d = a(n+1) - a(n). That works anywhere in the sequence, as long as the sequence is really arithmetic. If the differences change, then it is not an arithmetic sequence, even if the numbers look pattern-like at first glance.
Once you know d, you can use it to build the explicit formula for the nth term: a_n = a_1 + (n - 1)d. That formula is one of the main reasons common difference matters. It turns a pattern into a rule, so you can find the 50th term, the 100th term, or any other term without writing out the whole list.
Common difference also connects arithmetic sequences to straight-line thinking. The terms change at a constant rate, so if you graph term number versus term value, the points line up in a linear pattern. That is why arithmetic sequences feel so different from geometric sequences, where the change is multiplicative instead of additive.
Common difference is the piece that lets you move from a list of numbers to a usable algebraic rule. In College Algebra, that matters anytime a problem asks you to continue a sequence, identify the pattern, or write an nth-term formula.
It also shows up in word problems with steady change. A savings plan that adds the same amount each week, a stair-step pattern of values, or a repeated increase in measurements can often be modeled with an arithmetic sequence. If you can identify d, you can tell whether the pattern is growing or shrinking and how fast that change is happening.
Common difference is also the bridge to series. When you add the terms of an arithmetic sequence, the sum formulas use the first term, the number of terms, and d. So if you miss the common difference, you usually miss the easiest path to finding a sum.
A lot of mistakes in this unit come from treating every pattern like it is arithmetic. Checking the difference between consecutive terms is a quick filter. If the differences are not constant, you need a different model, not a forced arithmetic formula.
Keep studying College Algebra Unit 13
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view galleryArithmetic Sequence
A common difference only exists when the terms form an arithmetic sequence. If the gap between terms stays constant, the sequence is arithmetic, and d is the number that measures that repeated gap. If the gaps change from step to step, the sequence is not arithmetic, even if it still looks patterned.
a_n
The term a_n is the notation for the nth term of a sequence, and common difference is what lets you write an explicit formula for it. In arithmetic sequences, a_n = a_1 + (n - 1)d. That means d controls how each term depends on its position number.
Series
A series is the sum of the terms in a sequence, and common difference helps you add arithmetic sequences efficiently. Instead of adding terms one at a time, you can use arithmetic series formulas that depend on the first term and d. That is why finding d is often the first step.
Arithmetic Mean
Arithmetic mean is closely tied to arithmetic sequences because each middle term can act like a balanced step between neighbors. In a sequence with constant difference, a missing term can often be found by averaging the terms around it. That same constant spacing is what common difference measures.
A problem set question will usually give you a sequence and ask for the common difference, the next term, or the nth term. Your move is to subtract consecutive terms, check that the result stays the same, and then use that value in a formula like a_n = a_1 + (n - 1)d.
If the task asks for a missing term, you may need to work backward or forward by adding or subtracting d. If it asks whether a sequence is arithmetic, show the differences between terms instead of guessing from the shape of the numbers.
On quizzes, the most common trap is mixing up common difference with the actual terms. The difference is not the sequence itself, it is the constant change between terms. When a word problem describes equal increases or decreases over time, identify d first, then build the rule from there.
Common difference is the constant amount added or subtracted between consecutive terms in an arithmetic sequence.
You can find d by subtracting one term from the next term, and the result should stay the same throughout the sequence.
A positive common difference makes the sequence increase, while a negative common difference makes it decrease.
Once you know d, you can use it in the nth-term formula a_n = a_1 + (n - 1)d.
If the differences are not constant, the sequence is not arithmetic, even if the numbers look patterned.
Common difference is the constant number added or subtracted to move from one term of an arithmetic sequence to the next. In College Algebra, it is written as d. It is the feature that lets you identify arithmetic sequences and write formulas for them.
Subtract any term from the term that follows it: d = a(n+1) - a(n). For example, in 9, 13, 17, 21, the common difference is 4. If the subtraction gives the same result each time, the sequence is arithmetic.
No. The first term is the starting number in the sequence, while the common difference is the amount you add or subtract each time. In 6, 10, 14, 18, the first term is 6 and the common difference is 4.
You might be asked to identify the pattern, find a missing term, write an explicit formula, or decide whether a sequence is arithmetic. In each case, checking the difference between terms is usually the fastest first step. If the difference stays constant, you can use arithmetic sequence formulas.