Infinite geometric sequence
An infinite geometric sequence is a geometric pattern that continues forever, with each term found by multiplying the previous term by the same common ratio. In College Algebra, you use it to study repeated multiplication, long-term behavior, and geometric sums.
What is infinite geometric sequence?
An infinite geometric sequence in College Algebra is a sequence that keeps going without an ending term, and each new term is found by multiplying the previous one by the same common ratio. If the terms are written as a, ar, ar^2, ar^3, and so on, the sequence is geometric because the multiplier stays constant.
The general term is still written with the geometric sequence formula a_n = a_1 \cdot r^{n-1}. That formula does not change just because the sequence is infinite. What changes is that there is no last term to stop at, so you are usually interested in the pattern of the terms, whether they get bigger or smaller, and whether they approach a stable value.
The ratio r controls the behavior. If |r| < 1, the terms shrink in size as you move farther out in the sequence. For example, with a_1 = 8 and r = 1/2, the terms go 8, 4, 2, 1, 1/2, 1/4, and so on. The terms get closer and closer to 0, even though the sequence itself never stops.
If |r| > 1, the terms grow in magnitude, so the sequence diverges. That means the numbers do not settle down to a single value. A ratio like 3 makes the terms explode outward, while a negative ratio like -2 makes the sequence alternate signs and grow in size: 1, -2, 4, -8, 16, and so on.
A common point of confusion is mixing up the sequence with the sum of its terms. The sequence lists the terms one by one. The infinite geometric series asks for the sum of all those terms. Those are different ideas, and only the series has the formula S = a_1 / (1 - r), which works when |r| < 1.
So when you see an infinite geometric sequence in College Algebra, think pattern first: repeated multiplication, no final term, and behavior controlled by the ratio. Then use the nth-term formula or the sum formula only after you know which question you are actually being asked.
Why infinite geometric sequence matters in College Algebra
Infinite geometric sequences show up any time College Algebra turns repeated multiplication into a pattern you can analyze. They connect directly to exponential behavior, since multiplying by the same ratio over and over is the same basic idea behind growth and decay models.
This term matters because it gives you a clean way to describe a process that keeps going. A shrinking ratio models things like half-life style decay or discounting a value over and over, while a growing ratio matches repeated increases. Instead of listing an endless string of terms, you can write the pattern with a formula and predict where it is headed.
It also sets up infinite geometric series, which is where many class problems go next. Once you know when the terms shrink enough to settle, you can find a finite total for an endless pattern. That shows up in problem sets about savings plans, recurring payments, and approximation questions where you add more and more terms to get closer to the true total.
Another reason it matters is that it trains you to read the ratio correctly. A sign mistake or a wrong first term changes the whole sequence, so this is one of those topics where careful setup matters more than long calculation.
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open one-pagerHow infinite geometric sequence connects across the course
Geometric Sequence
An infinite geometric sequence is just a geometric sequence that never ends. The pattern rule is the same, but the infinite version makes you think about long-term behavior instead of stopping after a fixed number of terms.
Common Ratio
The common ratio is the number you multiply by each time, and it controls everything about the sequence. If you identify r correctly, you can write the terms, predict growth or decay, and decide whether the sequence converges or diverges.
Partial Sum
A partial sum adds only the first few terms of an infinite geometric sequence. In College Algebra, this is useful when you want an approximation or when you are checking whether the full infinite sum makes sense.
Divergent Sequence
When the terms of an infinite geometric sequence do not approach a single value, the sequence is divergent. This usually happens when the absolute value of the common ratio is greater than 1, or when the terms keep oscillating without settling.
Is infinite geometric sequence on the College Algebra exam?
A quiz or problem-set question usually gives you the first term and the common ratio, then asks for the nth term, the next term, or whether the sequence converges. Your job is to plug the values into a_n = a_1 \cdot r^{n-1}, then inspect r to see what happens as n gets large.
You may also be asked to decide whether the infinite geometric series has a sum. That means checking the size of r first, because the infinite sum only exists when |r| < 1. If the ratio is too large, too small, or equals 1 or -1 in a way that prevents settling, you should say the sequence diverges or the sum does not exist.
A common mistake is confusing the sequence terms with the sum of the terms. On a homework problem, write the terms if the prompt asks for the sequence, and use the sum formula only when it asks for the total of the infinite pattern.
Infinite geometric sequence vs Partial Sum
An infinite geometric sequence lists the terms in the pattern, while a partial sum adds only some of those terms. If a problem asks for the sequence, you do not total anything yet. If it asks for a partial sum, you stop at a chosen term and add only up to that point.
Key things to remember about infinite geometric sequence
An infinite geometric sequence is a geometric pattern that continues forever, with each term found by multiplying by the same common ratio.
The nth-term formula is a_n = a_1 \cdot r^{n-1}, so the first term and ratio tell you the whole sequence.
The size of r tells you what happens over time, because |r| < 1 makes the terms shrink and |r| > 1 makes them grow in magnitude.
Do not mix up the sequence with its sum, since the infinite sum uses a different formula and only works when |r| < 1.
In College Algebra, these sequences show up in repeated growth, decay, and any problem that uses repeated multiplication.
Frequently asked questions about infinite geometric sequence
What is an infinite geometric sequence in College Algebra?
It is a sequence that keeps going forever, and each term is found by multiplying the previous term by the same common ratio. You can write the terms with a_1, a_1r, a_1r^2, and so on. The key idea is that the pattern never ends, so you focus on the rule and the long-term behavior.
How do you know if an infinite geometric sequence converges?
Look at the common ratio. If |r| < 1, the terms get closer to 0, so the sequence converges to 0. If |r| > 1, the terms get larger in magnitude and the sequence diverges. If r is 1 or -1, the terms do not settle into a shrinking pattern.
What is the difference between an infinite geometric sequence and an infinite geometric series?
A sequence is the list of terms, while a series is the sum of those terms. For example, 3, 1.5, 0.75, ... is a sequence, but 3 + 1.5 + 0.75 + ... is a series. That difference matters because only the series uses the sum formula.
How do you find the nth term of an infinite geometric sequence?
Use a_n = a_1 \cdot r^{n-1}. Start with the first term, raise the common ratio to one less than the term number, and multiply. This is the fastest way to jump to any term without listing every term before it.