a_n is the formula label for the n-th term of an arithmetic sequence in Intermediate Algebra. It lets you find any term once you know the first term and common difference.
a_n is the notation for the n-th term of a sequence, and in Intermediate Algebra it usually shows up in arithmetic sequences. If you know the first term and the common difference, a_n tells you the exact value of the term in position n without listing every number before it.
For an arithmetic sequence, the terms change by the same amount each time. That constant change is the common difference, written as d. The explicit formula is a_n = a_1 + (n - 1)d, where a_1 is the first term. The n in a_n means the position in the sequence, not the value of the term itself.
That detail trips people up a lot. For example, if the sequence starts 4, 7, 10, 13, the common difference is 3, so a_1 = 4 and a_4 = 13. Plugging into the formula gives a_4 = 4 + (4 - 1)(3) = 13. The formula works because it counts how many jumps of size d happen after the first term.
You can use a_n to move forward or backward in the pattern. If you want the 20th term, you substitute n = 20. If you know a later term and the common difference, you can even work backward to find earlier terms or the first term. This is why the notation matters in algebra, it gives you a compact way to describe the whole sequence, not just one number.
In practice, a_n usually appears when a problem asks you to identify a term, write an explicit formula, or compare a sequence to a linear pattern. Because arithmetic sequences change by a constant amount, the graph of term number versus term value is linear. So a_n connects sequence notation to the linear thinking you already use in algebra.
a_n matters because it turns a repeating pattern into something you can calculate on demand. Instead of counting term by term, you can jump straight to any position in the sequence, which saves time and reduces mistakes on multi-step problems.
It also ties arithmetic sequences to linear patterns. In Intermediate Algebra, that connection comes up when you graph the sequence, look for a constant rate of change, or compare a table to an equation. The index n acts like the input, and the term value a_n acts like the output, so you get another way to see linear structure.
This term shows up in problems about saving money by the same amount each week, stair-step patterns in geometry, or any situation with a fixed increase or decrease. If the change is constant, a_n gives you a direct formula for prediction. If the change is not constant, then the sequence is not arithmetic and this formula will not fit.
Knowing a_n also helps you read notation correctly. Many mistakes come from mixing up the term number and the term value, or from forgetting that the first term uses n = 1, not n = 0. Once you can track that index, the rest of the sequence work becomes much cleaner.
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view galleryArithmetic Sequence
a_n is the label for one term inside an arithmetic sequence. If the sequence has a constant difference between terms, you can use the a_n formula to find any specific position without writing out the entire list. The notation only makes sense when you know the sequence is arithmetic.
Common Difference
The common difference is the amount added or subtracted each step, and it is the value that drives the change from one a_n term to the next. In the formula a_n = a_1 + (n - 1)d, d tells you how far the sequence moves from the first term. A wrong d gives a wrong term.
Explicit Formula
a_n is often found by an explicit formula because explicit formulas let you calculate any term directly from n. That is different from listing the sequence or building it step by step. For arithmetic sequences, the explicit formula is usually the fastest way to answer a question about a specific term.
Recursive Formula
A recursive formula finds a_n using earlier terms, while the explicit formula jumps straight to the term you want. Both describe the same arithmetic sequence, but they work differently. Recursive form is good for building the pattern, and a_n in explicit form is better for finding a far-away term quickly.
A quiz or problem set might give you a sequence and ask for a specific term like a_12 or a_25. Your job is to identify the first term, find the common difference, and plug the values into a_n = a_1 + (n - 1)d. You may also be asked to check whether a list of numbers is arithmetic by seeing whether the difference stays constant.
Another common task is translating a word problem into sequence notation. For example, if a savings plan adds the same amount each week, you can model the weekly balance with a_n and then find the balance after a given number of weeks. If the prompt gives a term value and asks for the position, you may need to solve the formula for n instead of just evaluating it.
a_n is often confused with a recursive formula because both describe sequence terms. The difference is that a_n usually refers to the explicit n-th term formula, which lets you jump straight to any term. A recursive formula depends on earlier terms, so you have to build the sequence step by step.
a_n means the n-th term of a sequence, and in Intermediate Algebra it most often refers to an arithmetic sequence.
The explicit arithmetic sequence formula is a_n = a_1 + (n - 1)d, where a_1 is the first term and d is the common difference.
The n in a_n tells you the term position, so a_1 is the first term, a_2 is the second term, and so on.
If a sequence changes by the same amount each time, you can use a_n to find any term without listing every previous term.
A common mistake is confusing the term number with the term value or using the wrong difference.
a_n is the notation for the n-th term of a sequence, usually an arithmetic sequence in Intermediate Algebra. It names the term in position n, so a_5 means the fifth term, a_12 means the twelfth term, and so on. You usually find it with the explicit formula a_n = a_1 + (n - 1)d.
First find the first term a_1 and the common difference d. Then substitute the term number n into a_n = a_1 + (n - 1)d. If the sequence is 6, 9, 12, 15, then d = 3 and a_4 = 6 + 3(3) = 15.
No. a_n is the value of a specific term, while the common difference is the constant amount added or subtracted between terms. In an arithmetic sequence, the common difference is usually written as d. The formula uses both, but they are not the same thing.
Yes. If a situation changes by a constant amount each step, you can model it with an arithmetic sequence and use a_n to predict a future value. That shows up in savings plans, stair-step patterns, schedules, and other linear growth or decline problems.