A finite sequence is an ordered list in College Algebra that has a set number of terms. Each term has a position, like first, second, or nth, and the list ends after a fixed last term.
A finite sequence in College Algebra is an ordered list of terms with a fixed number of entries. The order matters, so 2, 4, 6 is different from 6, 4, 2 even though they use the same numbers.
You usually write a finite sequence as a list like a1, a2, ..., an, or with set-style notation such as {ai} from i = 1 to n. The notation tells you two things at once: what each term is called and how many terms there are. That fixed last index, n, is what makes the sequence finite.
A finite sequence can be defined by a pattern instead of listing every term. For example, an arithmetic sequence adds the same amount each time, while a geometric sequence multiplies by the same number each time. In either case, the sequence ends after a chosen number of terms, so you can talk about the first term, the last term, and any term in between.
One common trap is mixing up a sequence and a series. A sequence is the list itself, while a series is the sum of the terms in that list. So the finite sequence 3, 6, 9, 12 becomes the series 3 + 6 + 9 + 12 if you are adding the terms.
In College Algebra, finite sequences show up when you need to identify a pattern, find an nth term, or use a formula to generate terms without writing them all out. The fixed length makes them easier to count, compare, and sum than an infinite sequence.
Finite sequences are the setup for a lot of sequence and series work in College Algebra. If you can tell where the list starts, how it changes, and where it ends, you can decide whether a pattern is arithmetic, geometric, or something else.
That matters because many algebra problems ask you to move between a written pattern and a formula. You might be given a list of terms and asked for the next term, the nth term, or the number of terms in the list. A finite sequence gives you a structure for all of those moves.
It also matters when the class shifts from listing terms to adding them. Once the sequence is finite, you can form a series and use sum formulas instead of adding each term by hand. That is a big shortcut for arithmetic and geometric sums.
A lot of mistakes come from ignoring position. In a finite sequence, the term number is part of the meaning, so a2 is not just another value, it is the second value in the pattern. That makes indexing and notation a big part of the skill.
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view galleryArithmetic Sequence
An arithmetic sequence is one common type of finite sequence in College Algebra. The terms change by adding the same constant difference each time, so the pattern is easy to track once you know the first term and the common difference. Many finite-sequence problems ask you to spot this pattern, write an explicit formula, or list a specific number of terms.
Geometric Sequence
A geometric sequence is another major finite sequence pattern. Instead of adding the same amount, each term is found by multiplying by the same ratio. That makes the sequence grow or shrink faster than an arithmetic sequence, and it often shows up in formulas, repeated growth, and series questions.
Series
A series is what you get when you add the terms of a finite sequence. This is the next step after identifying the sequence itself, because now you are working with the total rather than the ordered list. In class problems, the sequence may be given first, then you are asked to find the sum of its terms.
A quiz or problem-set question usually gives you a list of terms, a pattern, or a formula and asks you to identify whether the sequence is finite, find the number of terms, or write the next few terms. You may also be asked to tell whether it is arithmetic or geometric, since that changes which formula you use.
If the question moves to summing, you first decide whether you are dealing with a sequence or a series. Then you use the finite nature of the list to count terms, plug into a formula, or check the last index. A common scoring mistake is dropping the order or forgetting that the final term is part of the setup.
A finite sequence is the ordered list of terms, while a series is the sum of those terms. If you are naming or generating the terms, you are working with a sequence. If you are adding them, you are working with a series.
A finite sequence is an ordered list with a fixed number of terms.
The position of each term matters, so the first term, second term, and last term are all different parts of the pattern.
You can write a finite sequence by listing terms or by using notation like a1 through an.
Arithmetic and geometric sequences are two common finite sequence types in College Algebra.
A sequence lists the terms, but a series adds them.
A finite sequence in College Algebra is an ordered list with a set number of terms. The sequence starts at a first term and ends at a last term, so it does not continue forever. You can describe it by listing terms or by giving a rule for the pattern.
A finite sequence is the list of terms, and a series is the sum of those terms. For example, 2, 5, 8, 11 is a sequence, while 2 + 5 + 8 + 11 is a series. That difference matters a lot when a problem asks for a sum instead of the pattern itself.
A sequence is finite if it has a definite last term or a fixed number of terms. If the problem gives you a list that stops, or says there are n terms, it is finite. If the terms keep going without an end, it is infinite.
An arithmetic sequence like 4, 7, 10, 13 is finite if it stops there, and a geometric sequence like 3, 6, 12, 24 is also finite if the list ends. The same pattern can be finite or infinite depending on whether the terms stop at a chosen point.