A 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.
Each term is the one before it times the same number. With r = 2 the bars double; with r = 12 the terms halve.
Formal definition and notation
A sequence a1, a2, a3, … is geometric if there is a constant r ≠ 0 with an = r · an − 1 for every n ≥ 2; equivalently, anan − 1 = r for all n ≥ 2 (with a1 ≠ 0). Its explicit formula is an = a1rn − 1, an exponential function of n restricted to whole-number inputs.
Symbol
Read it as
Meaning
an
a sub n
the nth term
a1
a sub one
the first term
r
r
the common ratio: any term divided by the term before it
an = r · an − 1
a sub n equals r times a sub n minus 1
the recursive rule: multiply the previous term by r
Key facts
Test: divide each term by the one before it. If every quotient is the same, the sequence is geometric.
r > 1: growth (2, 6, 18, …). 0 < r < 1: decay (80, 40, 20, …). r < 0: the signs alternate (2, −6, 18, −54, …).
Repeated percent change is geometric: growing 5% a year multiplies by r = 1.05 each year.
An arithmetic sequence adds the same number each step; a geometric sequence multiplies by the same number.
Sum of the first n terms (for r ≠ 1): Sn = a1(1 − rn)1 − r.
Geometric sequence formula
Geometric sequence formula
an = a1 · rn − 1
an
the term you want, in position n
a1
the first term
r
the common ratio, r = a2a1
n − 1
the number of multiplications from a1 to an
The exponent is n − 1, not n: to reach the 5th term you multiply by r only 4 times.
Try it: multiply by the ratio
Interactive
The machine multiplies by the common ratio 3. Type the last output back in as the next input and press Run to build the sequence 2, 6, 18, 54 … one term at a time.
This interactive needs JavaScript. The worked examples below show the same ideas step by step.
Geometric Sequence examples, step by step
Example 1: Find the ratio and the next term
Warm-up
Find the common ratio and the next term: 5, 15, 45, 135, …
Divide neighbors: 155 = 3, 4515 = 3, 13545 = 3. So r = 3.
Next term: 135 × 3 = 405.
Answer: The common ratio is 3 and the next term is 405.
Example 2: A far-away term
Core
Find the 8th term of 64, 32, 16, 8, …
a1 = 64 and r = 3264 = 12.
Use an = a1rn − 1 with n = 8: a8 = 64 · (12)7.
(12)7 = 1128.
a8 = 64128 = 12.
Answer: The 8th term is 12, or 0.5.
Example 3: From two terms
Test-style
In a geometric sequence, a2 = 12 and a5 = 324. Find r and a1.
From term 2 to term 5 you multiply by r three times: 12 r3 = 324.
r3 = 27, so r = 3.
a1 = a2r = 123 = 4.
Check: 4, 12, 36, 108, 324.
Answer:r = 3 and a1 = 4.
Common mistakes
✗ MistakeFor 4, 12, 36, …, a6 = 4 · 36.
✓ Insteada6 = 4 · 35 = 972.
The first term already counts as term 1, so only n − 1 = 5multiplications reach term 6.
✗ Mistake4 · 35 = 125
✓ Instead4 · 35 = 4 · 243 = 972
Order of operations: the exponent applies only to r. Raise 3 to the 5th power first, then multiply by 4.
An arithmetic sequence adds a common difference (5, 8, 11, 14 adds 3); a geometric sequence multiplies by a common ratio (5, 15, 45, 135 multiplies by 3).
The common difference is the number added in an arithmetic sequence. A geometric sequence has a common ratio instead: the number you multiply by.
Geometric Sequencevs.Exponential function
f(x) = a · bx is defined for every real x. A geometric sequence uses only whole-number positions, so its graph is a set of separate dots.
Quick check: 3 questions
Question 1 of 3
Which sequence is geometric?
Show answer and explanation
Answer: B. 2, 6, 18, 54 multiplies by 3 every time. 3, 7, 11, 15 adds 4 (arithmetic), 1, 4, 9, 16 are squares, and 5, 10, 20, 25 doubles twice and then adds 5.
Question 2 of 3
What is the 6th term of 1, −2, 4, −8, …?
Show answer and explanation
Answer: B.r = −2, so a6 = 1 · (−2)5 = −32. An odd power of a negative is negative; 32 drops the sign, 64 uses the 7th term, and −16 is the 5th term with the wrong sign.
Question 3 of 3
A town of 10,000 grows 4% each year. Which expression gives the population after n years?
Show answer and explanation
Answer: D. Growing 4% multiplies by 1 + 0.04 = 1.04 each year, so the populations form a geometric sequence with ratio 1.04. 10,000 + 400n adds the same amount each year, which is arithmetic, and 0.04n would shrink toward 0.
Geometric Sequence: frequently asked questions
How do you know if a sequence is geometric?
Divide each term by the term before it. If you get the same number every time, the sequence is geometric and that number is the common ratio. For 7, 14, 28, 56 every quotient is 2. If the differences are equal instead, the sequence is arithmetic. Check all the pairs, not just the first one.
Can the common ratio be negative or a fraction?
Yes. A negative ratio makes the terms alternate signs: 3, −6, 12, −24 has r = −2. A ratio between 0 and 1 makes the terms shrink toward 0: 100, 25, 6.25 has r = 14. The only ratio not allowed is 0, because then every term after the first would be 0.
What is the difference between a geometric sequence and a geometric series?
A sequence is the list of terms: 2, 6, 18, 54. A series is their sum: 2 + 6 + 18 + 54 = 80. For a geometric series the sum of the first n terms is Sn = a1(1 − rn)1 − r. Here that gives 2(1 − 81)1 − 3 = 80.
Updated November 2026·Reviewed by the Effortless Math team·Standards: CCSS HSF-BF.A.2, CCSS HSF-LE.A.2, CCSS HSF-IF.A.3
Cite this entry
Effortless Math Team. “Geometric Sequence.” Effortless Math Dictionary. Updated November 2026. https://www.effortlessmath.com/math-dictionary/geometric-sequence/
Flashcard
TermGeometric Sequencean = a1 · rn − 1
DefinitionA list where each term is the previous term times the same ratio r. 3, 6, 12, 24 has r = 2.
ExponentAn exponent tells how many times to use a base as a factor. It is the small raised number to the upper right of the base: in 53, the base is 5 and the exponent is 3, so 53 = 5 × 5 × 5 = 125. Any nonzero number with an exponent of 0 equals 1.Algebra · Grade 6, Grade 7, Grade 8Common RatioThe 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.Functions · Algebra 1, Algebra 2QuotientA 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 5Percent ChangePercent change measures how much a value grew or shrank relative to where it started: the difference divided by the original amount, written as a percent. From 50 to 60 is a 20% increase; from 80 to 60 is a 25% decrease. The original value is always the base, which is the step people get wrong.Ratios and Proportions · Grade 7, Grade 8, Algebra 1Arithmetic 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 2NumberA 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 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 4Order of OperationsThe order of operations is the agreed sequence for evaluating an expression: parentheses first, then exponents, then multiplication and division from left to right, then addition and subtraction from left to right. Without it the same expression would have several answers. Many students remember it as PEMDAS.Algebra · Grade 5, Grade 6, Grade 7PowerA power is a number written as a base with an exponent, such as 53, or the value that results, 53 = 5 × 5 × 5 = 125. We say 125 is the third power of 5, or “5 to the third power”. The second power is called the square and the third power the cube.Number System · Grade 5, Grade 6, Grade 7RatioA 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 8DifferenceA 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 3Common 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 2SeriesA 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, CalculusMultipleA multiple of a number is the result of multiplying that number by any whole number. The multiples of 4 are 4, 8, 12, 16, and so on, continuing without end. Every number is a multiple of itself and of 1. To test a number, divide: if the remainder is 0, it is a multiple.Number System · Grade 4, Grade 5, Grade 6SequenceA 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 2Exponential 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, PrecalculusFunctionA function is a rule that gives exactly one output for each input. If you put in one x-value, a function gives back one y-value. In f(x) = 2x + 1, the input 3 gives the output 7. Two inputs may share an output, but one input can never have two different outputs.Functions · Grade 8, Algebra 1, Algebra 2