The common ratio is the constant multiplier between consecutive terms in a geometric sequence. In Intermediate Algebra, you use it to identify geometric patterns and find later terms.
In Intermediate Algebra, the common ratio is the number you multiply by each time to get from one term of a geometric sequence to the next. It is usually written as r, and it stays the same across the whole sequence.
For example, in 3, 6, 12, 24, each term is found by multiplying the previous term by 2. That makes the common ratio 2. In 81, 27, 9, 3, each term is found by multiplying by 1/3, so the common ratio is 1/3.
This is what makes a sequence geometric instead of arithmetic. Arithmetic sequences change by adding or subtracting the same amount, while geometric sequences change by multiplying by the same amount. That difference matters because multiplication creates curved growth or decay, not straight-line change.
A common ratio can be greater than 1, between 0 and 1, or negative. If r is greater than 1, the terms grow larger each step. If 0 < r < 1, the terms get smaller and the sequence shows decay. If r is negative, the signs alternate, so the sequence flips back and forth between positive and negative values.
You find the common ratio by dividing a term by the term before it, as long as the previous term is not zero. For instance, in a sequence like 5, 20, 80, you check 20/5 = 4 and 80/20 = 4, so r = 4. If the quotients do not match, the sequence is not geometric.
Once you know r, you can continue the pattern, check whether a sequence is geometric, and build formulas for future terms. That makes common ratio one of the main tools for working with sequences in this course.
The common ratio is the piece that turns a list of numbers into a pattern you can actually predict. In Intermediate Algebra, that means you can tell whether a sequence is geometric, decide if it is growing or shrinking, and extend it without guessing.
It also connects directly to exponential behavior. When each term is multiplied by the same factor, the change is multiplicative, not additive, so the numbers can rise very quickly or shrink toward zero. That shows up in topics like exponential growth and decay, which are built on the same repeated-multiplication idea.
This term also helps you work backward. If you know the pattern and one term, you can often recover earlier or later terms, check whether a rule fits, or compare two sequences. That kind of reasoning shows up in homework problems, quizzes, and word problems where the sequence is embedded in a table or story.
A lot of mistakes come from mixing up multiplication and addition. If a sequence changes by the same difference each time, you are looking at a common difference, not a common ratio. Spotting that distinction saves time and keeps you from using the wrong formula.
Keep studying Intermediate Algebra Unit 12
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view galleryGeometric Sequence
A geometric sequence is the larger pattern that uses a common ratio. If every term is found by multiplying by the same number, the sequence is geometric. Identifying the ratio is usually the first step before you write a rule or find later terms.
Arithmetic Sequence
Arithmetic sequences change by a constant difference, not a multiplier. Students often confuse the two because both have a repeating pattern, but the operation matters. If you are adding the same amount each time, you need common difference, not common ratio.
Exponential Growth
A common ratio greater than 1 creates exponential growth because each term gets multiplied by the same factor. That is why geometric sequences are a basic model for things like population increases or money growing with repeated interest.
Explicit Formula
Once you know the common ratio, you can often write an explicit formula for the nth term of a geometric sequence. Instead of listing every step, the formula lets you jump straight to any term position, which is useful on problems with large term numbers.
A problem set or quiz usually asks you to find the common ratio from a list, table, or word problem, then use it to generate more terms. You might divide consecutive terms, check whether a pattern is geometric, or decide whether the sequence shows growth or decay. If the numbers are messy, you may need to simplify fractions carefully and test more than one pair of terms. Another common task is to compare a geometric sequence with an arithmetic one, so you need to tell whether the pattern is multiplicative or additive. In a written response, you may also explain why a sequence is or is not geometric by showing the quotients between terms.
Common ratio and common difference are easy to mix up because both describe patterns between terms. Common ratio means you multiply by the same number each time, while common difference means you add or subtract the same number each time. If the sequence is 2, 6, 18, 54, the ratio is 3. If it is 2, 5, 8, 11, the difference is 3.
The common ratio is the constant multiplier between consecutive terms in a geometric sequence.
You find it by dividing a term by the term right before it, and the result should stay the same throughout the sequence.
A ratio greater than 1 means growth, while a ratio between 0 and 1 means decay.
If the terms change by adding the same amount instead of multiplying, the sequence is arithmetic, not geometric.
Knowing the common ratio lets you extend a pattern and write formulas for future terms.
The common ratio is the number you multiply by to move from one term in a geometric sequence to the next. In Intermediate Algebra, it tells you whether the pattern is growing, shrinking, or alternating. You usually find it by dividing consecutive terms.
Divide any term by the term before it. For 4, 12, 36, the ratio is 12/4 = 3 and 36/12 = 3, so the common ratio is 3. If the quotients do not match, the sequence is not geometric.
Common ratio means multiply by the same number each step, while common difference means add the same number each step. Ratio goes with geometric sequences, and difference goes with arithmetic sequences. That difference changes the whole pattern and the formulas you use.
Once you know the ratio, you can generate later terms by repeated multiplication. You can also check whether a pattern is geometric or use it to build an explicit formula. If the ratio is negative, remember that the signs will alternate.