The common ratio is the fixed number you multiply each term of a geometric sequence by to get the next term. Find it by dividing any term by the term before it: r = anan − 1. In 5, 20, 80, 320, the common ratio is 4. It can be a fraction or negative, but never 0.
In Spanish
razón común
Also called
ratio of a geometric sequence, constant ratio, common multiplier, r
The same multiplier links every pair of neighbors in a row. A ratio above 1 makes terms grow, a ratio between 0 and 1 makes them shrink, and a negative ratio flips the sign each time.
Formal definition and notation
A sequence a1, a2, a3, … of nonzero terms is geometric if there is a constant r ≠ 0 with an = r · an − 1 for every n ≥ 2. This constant r = anan − 1 is the common ratio. The explicit formula is an = a1rn − 1.
Symbol
Read it as
Meaning
r
r
the common ratio
anan − 1
a sub n over a sub n minus 1
a term divided by the term before it
an = a1rn − 1
a sub n equals a sub 1 times r to the n minus 1
the nth term of a geometric sequence
Key facts
r > 1: terms grow (2, 6, 18). 0 < r < 1: terms shrink toward 0 (100, 50, 25).
r < 0: the signs alternate (3, −6, 12, −24).
The common ratio plays the role of the growth factor b in an exponential function.
If two terms are k places apart, their quotient is rk: a5a2 = r3.
An infinite geometric series has a sum only when |r| < 1.
A ratio compares any two quantities, like 3 boys to 5 girls. A common ratio is one specific multiplier shared by every pair of neighbors in a geometric sequence.
Quick check: 3 questions
Question 1 of 3
What is the common ratio of 7, 21, 63, 189?
Show answer and explanation
Answer: D.217 = 3, and every other pair also gives 3. 14 is the difference 21 − 7, one third divides the wrong way, and 7 is the first term.
Question 2 of 3
Which sequence has a common ratio of −12?
Show answer and explanation
Answer: B.−48 = −12 and the signs alternate. The other halving sequences have ratio 12, and 8, 6, 4, 2 subtracts 2.
Question 3 of 3
A geometric sequence has a2 = 10 and a4 = 90, with all terms positive. What is r?
Show answer and explanation
Answer: A.a4 = a2r2, so r2 = 9 and r = 3. 9 is r2, 40 is half the difference, and 4.5 divides 9 by 2 instead of taking a square root.
Common Ratio: frequently asked questions
How do you find the common ratio?
Divide any term by the term just before it: r = a2a1. Do it for a second pair to make sure the sequence really is geometric. For 6, 18, 54, both 186 and 5418 give 3. If you only know two terms that are far apart, use aman = rm − n and take a root.
Can the common ratio be negative or a fraction?
Yes to both. A negative ratio makes the signs alternate, as in 2, −6, 18 with r = −3. A fraction between 0 and 1 makes the terms shrink, as in 64, 16, 4 with r = 14. The ratio cannot be 0, because then every term after the first would be 0 and dividing would fail.
What is the difference between a common ratio and a common difference?
A common difference is the number you add to each term of an arithmetic sequence; find it by subtracting. A common ratio is the number you multiply by in a geometric sequence; find it by dividing. In 2, 4, 6, 8 the difference is 2; in 2, 4, 8, 16 the ratio is 2.
Geometric SequenceA geometric sequence is a list of numbers where each term is the term before it times the same number, called the common ratio r. In 3, 6, 12, 24, … the ratio is 2. The nth term is an = a1 · rn − 1. The terms grow when |r| > 1 and shrink when |r| < 1.Functions · Algebra 1, Algebra 2, PrecalculusExponential FunctionAn exponential function is a function in which the input is the exponent: f(x) = a · bx, with a ≠ 0, base b > 0 and b ≠ 1. Each time x goes up by 1, the output is multiplied by the same factor b. If b > 1 the function grows; if 0 < b < 1 it decays.Functions · Algebra 1, Algebra 2, PrecalculusQuotientA quotient is the answer you get when you divide one number by another. In 20 ÷ 4 = 5, the quotient is 5. The number being divided is the dividend and the number you divide by is the divisor. When the division is not exact, the quotient comes with a remainder, or as a decimal or fraction.Operations · Grade 3, Grade 4, Grade 5SeriesA series is the sum of the terms of a sequence. The sequence 3, 7, 11, 15 is a list; the series 3 + 7 + 11 + 15 = 36 is what you get by adding it. A finite series stops after n terms, written Sn. An infinite series has a sum only if its partial sums approach a fixed number.Functions · Algebra 2, Precalculus, CalculusCommon DifferenceThe common difference is the fixed amount added to each term of an arithmetic sequence to get the next term, written d. Find it by subtracting any term from the term after it: in 7, 11, 15, 19, d = 11 − 7 = 4. A positive d makes the terms increase, a negative d makes them decrease.Functions · Grade 8, Algebra 1, Algebra 2Arithmetic SequenceAn arithmetic sequence is a list of numbers in which you add the same amount each time to get the next term. That fixed amount is the common difference, d. In 4, 7, 10, 13, … the common difference is 3. The nth term is an = a1 + (n − 1)d.Functions · Grade 8, Algebra 1, Algebra 2MultiplicationMultiplication is the operation that combines equal groups to find a total. The fact 4 × 6 = 24 means 4 groups of 6, or 6 + 6 + 6 + 6. The numbers you multiply are called factors, and the result is the product. Order does not matter: 6 × 4 is also 24.Operations · Grade 2, Grade 3, Grade 4Cube RootThe cube root of a number is the value that, used as a factor three times, gives that number. It is written 3√x. Because 4 × 4 × 4 = 64, 3√64 = 4. Unlike square roots, cube roots of negative numbers are real numbers: 3√−27 = −3.Number System · Grade 8, Algebra 1, Algebra 2RatioA ratio compares two quantities by division. Ratios can be written with words, a colon, or a fraction: 3 boys to 5 girls is 3 to 5, 3 : 5, or 35. Order matters, and a ratio can compare part to part or part to whole. Multiplying both numbers by the same amount gives an equivalent ratio.Ratios and Proportions · Grade 6, Grade 7, Grade 8NumberA number is an amount or value used to count, measure, order and compare. We write numbers with digits (7), words (seven) or other symbols, but the number itself is the amount: 7 dots, the word seven and the numeral 7 all show the same number. Numbers include whole numbers, fractions, decimals and negatives.Number System · Kindergarten, Grade 1, Grade 2DifferenceA difference is the result of subtracting one number from another. In 12 − 7 = 5, the difference is 5. It tells how much bigger one number is than another, or how far apart they are on a number line. “The difference of a and b” means a − b, in that order.Operations · Grade 1, Grade 2, Grade 3SequenceA sequence is an ordered list of numbers that follows a rule, such as 3, 7, 11, 15, …. Each number is a term, and its position is 1st, 2nd, 3rd and so on. You can describe a sequence with a formula for the nth term, an, or by how each term comes from the one before it.Functions · Grade 8, Algebra 1, Algebra 2Recursive FormulaA recursive formula defines a sequence by giving its first term and a rule that builds each next term from the term or terms before it. For example, a1 = 5 and an = an − 1 + 4 make 5, 9, 13, 17. To find a later term, you compute every term before it, one step at a time.Functions · Algebra 1, Algebra 2Exponential GrowthExponential growth is an increase by the same percent, or the same factor, in each equal time period, so the amount grows faster and faster. It is modeled by y = a(1 + r)t, where a is the starting amount and r is the growth rate. Growing 10% a year multiplies the amount by 1.1 every year.Functions · Algebra 1, Algebra 2