A 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.
A sequence pairs each position with a term. The recursive rule looks back one step (+4); the explicit rule goes straight from n to the term. Because n is a counting number, the graph is dots, not a line.
Formal definition and notation
A sequence is a function whose domain is the positive integers1, 2, 3, … (some books start at 0). The output for position n is the nth term, written an. A finite sequence has a last term; an infinite sequence goes on forever, shown by three dots. An explicit formula gives an directly from n. A recursive formula gives the first term and a rule that builds each term from the term or terms before it.
Symbol
Read it as
Meaning
an
a sub n
the nth term, the term in position n
a1
a sub 1
the first term
an − 1
a sub n minus 1
the term just before an; used in recursive formulas
…
and so on
the pattern continues forever (an infinite sequence)
Other patterns are sequences too: the squares 1, 4, 9, 16, … follow an = n2, and in 1, 1, 2, 3, 5, 8, … (the Fibonacci sequence) each term is the sum of the two before it.
The graph of a sequence is a set of separate dots, not a connected line, because the position n can only be 1, 2, 3 and so on.
An arithmetic sequence follows a linear pattern and a geometric sequence follows an exponential pattern.
Adding the terms of a sequence gives a series: the series for 3, 7, 11, 15 is 3 + 7 + 11 + 15 = 36.
Two ways to write a sequence rule
Two ways to write a sequence rule
explicit: an = 4n − 1; recursive: a1 = 3, an = an − 1 + 4
n
the position: 1, 2, 3, …
an
the term in position n
an − 1
the previous term
Both rules give 3, 7, 11, 15, …. The explicit rule jumps straight to any term: a50 = 4(50) − 1 = 199. The recursive rule needs every term before it, so it is better for describing step-by-step growth.
Try it: position in, term out
Interactive
Here the machine is the explicit rule of the square numbers, a sub n = n squared. Type a position such as 5, 6 or 10 and press Run to get that term. Notice the gaps between terms, 3, 5, 7, …, are not equal, so this sequence is not arithmetic.
This interactive needs JavaScript. The worked examples below show the same ideas step by step.
Sequence examples, step by step
Example 1: Continue the pattern
Warm-up
Find the next two terms of 50, 44, 38, 32, ….
Find the change between terms: 44 − 50 = −6, 38 − 44 = −6, 32 − 38 = −6.
The rule is subtract 6, so the next term is 32 − 6 = 26.
Then 26 − 6 = 20.
Answer: The next two terms are 26 and 20.
Example 2: Write the nth term and use it
Core
Write an explicit formula for 5, 8, 11, 14, … and find the 40th term.
The terms go up by 3 each time, so the formula has 3n.
At n = 1, 3n = 3, but the first term is 5, so add 2: an = 3n + 2.
Check: a2 = 3(2) + 2 = 8. Correct.
a40 = 3(40) + 2 = 122.
Answer:an = 3n + 2, and the 40th term is 122.
Example 3: Use a recursive formula
Test-style
A sequence is defined by a1 = 2 and an = 3an − 1 − 1. What is a4?
a2 = 3(2) − 1 = 5.
a3 = 3(5) − 1 = 14.
a4 = 3(14) − 1 = 41.
Answer:a4 = 41.
Common mistakes
✗ MistakeFor 3, 7, 11, …, saying a2 = 2.
✓ Insteada2 is the 2nd term, so a2 = 7.
The small number n is the position. The term is the value found at that position.
✗ MistakeFor 5, 8, 11, …, writing an = 3n + 5.
✓ InsteadTest n = 1: 3(1) + 5 = 8, not 5. The correct rule is an = 3n + 2.
The first term matches n = 1, not n = 0. Always check your formula against the first term.
✗ MistakeAssuming every sequence adds: for 2, 6, 18, …, giving 30 next (+12).
✓ Instead6 ÷ 2 = 3 and 18 ÷ 6 = 3, so the rule is multiply by 3. The next term is 54.
Check both the differences and the ratios before you decide on a rule.
✗ MistakeJoining the points of a sequence graph with a solid line.
✓ InsteadPlot separate dots at n = 1, 2, 3, …, with nothing in between.
A sequence has no 1.5th term, so the points between the dots are not part of it.
Not to be confused with
Sequencevs.Series
A series is the sum of the terms of a sequence. The sequence 2, 4, 6, 8 is a list; the series 2 + 4 + 6 + 8 is one number, 20.
Sequencevs.Set
In a set, order and repeats do not matter: {1, 2, 3} is the same set as {3, 2, 1}. In a sequence, 1, 2, 3 and 3, 2, 1 are different, and terms can repeat.
An arithmetic sequence is one kind of sequence, the kind that adds the same number every time. 1, 4, 9, 16 is a sequence, but not an arithmetic one.
Quick check: 3 questions
Question 1 of 3
What is the 5th term of an = 2n + 3?
Show answer and explanation
Answer: C. Substitute n = 5: 2(5) + 3 = 13. 10 forgets the +3, 11 uses n = 4, and 25 multiplies 2 + 3 by 5.
Question 2 of 3
What is the next term of 1, 4, 9, 16, …?
Show answer and explanation
Answer: B. These are the square numbers, 12, 22, 32, 42, so next is 52 = 25. The gaps 3, 5, 7 grow by 2, so the next gap is 9. 23 repeats the gap 7, 20 adds 4, and 32 doubles.
Question 3 of 3
A sequence has a1 = 10 and an = an − 1 − 4. Which explicit formula gives the same terms?
Show answer and explanation
Answer: D. The terms are 10, 6, 2, …, and 14 − 4(1) = 10, 14 − 4(2) = 6. 10 − 4n starts at 6, 4n + 6 goes up instead of down, and 10n − 4 uses the wrong rate.
Sequence: frequently asked questions
What is the difference between a sequence and a series?
A sequence is a list of numbers in order, such as 1, 3, 5, 7. A series is what you get when you add those numbers: 1 + 3 + 5 + 7 = 16. So a sequence gives many terms, while a finite series gives one total. Sigma notation is the usual shorthand for writing a series.
How do you find the nth term of a sequence?
First find the pattern. If the same number d is added each time, the nth term is an = a1 + (n − 1)d. If each term is multiplied by the same number r, it is an = a1 · rn − 1. For other patterns, compare each term with its position, as in 1, 4, 9, 16, which is n2.
Is a sequence a function?
Yes. A sequence matches each position n = 1, 2, 3, … with exactly one term, which is what a function does. The input is the position and the output is the term. That is why its graph is a set of dots above the counting numbers, and why an is sometimes written f(n).
FunctionA 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 2DomainThe domain of a function is the set of all input values (usually x-values) the function can accept. Unless a problem says otherwise, it is every real number except values that would divide by zero or take an even root of a negative number. On a graph, the domain is how far the graph reaches left and right.Functions · Grade 8, Algebra 1, Algebra 2Positive IntegerA positive integer is a whole number greater than zero: 1, 2, 3, 4 and so on. Positive integers have no fraction or decimal part and are never 0 or negative. They are also called counting numbers, and the set is written ℤ+. When a test says n is a positive integer, n cannot be 0, −3 or 2.5.Problem Solving · Grade 6, Grade 7, Algebra 1OutputAn output is the value a function or rule produces from a given input. You put a number in, apply the rule, and the result that comes out is the output. For y = 2x, the input 3 gives the output 6. In an input-output table the output sits in the second column.Functions · Grade 6, Grade 7, Grade 8Recursive 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 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 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 2Common 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 2Geometric 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, PrecalculusCommon 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 2SquareA square is a quadrilateral with four equal sides and four right angles. It is a special rectangle and a special rhombus at the same time. For a square with side s, the perimeter is 4s, the area is s2, and each diagonal is s√2.Geometry · Kindergarten, Grade 2, Grade 3Linear EquationA linear equation is an equation whose graph is a straight line. In one variable, it usually has a variable to the first power, like 3x + 2 = 14. In two variables, it can be written in forms like y = mx + b, where m is the slope. No variable is squared, multiplied by another variable or in a denominator.Algebra · Grade 7, Grade 8, Algebra 1ExponentAn 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 8SeriesA 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, CalculusDifferenceA 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 3RatioA 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 8InputAn input is the value you put into a function or a rule. The rule acts on the input to produce the output: for f(x) = x + 2, the input 5 gives the output 7. For many functions, the input is the x-value. In an input-output table, the inputs fill the first column.Functions · Grade 6, Grade 7, Grade 8Linear FunctionA linear function is a function whose graph is a straight line. You can write it as f(x) = mx + b, where m is the slope, or constant rate of change, and b is the y-intercept. Each time x increases by 1, the output changes by the same amount, m.Functions · Grade 8, Algebra 1