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Sequence — Definition, Formula & Examples

Sequence

A list of numbers set apart by commas, such as 1, 3, 5, 7, . . .

 

 

See also

Series, arithmetic sequence, geometric sequence

Key Formula

a1,  a2,  a3,  …,  an,  …a_1,\; a_2,\; a_3,\; \ldots,\; a_n,\; \ldots
Where:
  • ana_n = The nth term (general term) of the sequence
  • nn = The position of the term, where n is a positive integer (1, 2, 3, …)

Worked Example

Problem: A sequence is defined by the rule a_n = 3n + 1. Find the first five terms.
Step 1: Substitute n = 1 into the rule to find the first term.
a1=3(1)+1=4a_1 = 3(1) + 1 = 4
Step 2: Substitute n = 2 to find the second term.
a2=3(2)+1=7a_2 = 3(2) + 1 = 7
Step 3: Substitute n = 3 to find the third term.
a3=3(3)+1=10a_3 = 3(3) + 1 = 10
Step 4: Continue with n = 4 and n = 5.
a4=3(4)+1=13,a5=3(5)+1=16a_4 = 3(4) + 1 = 13, \quad a_5 = 3(5) + 1 = 16
Answer: The first five terms of the sequence are 4, 7, 10, 13, 16.

Another Example

Problem: Find the 10th term of the sequence 2, 6, 18, 54, … where each term is 3 times the previous term.
Step 1: Identify the pattern. Each term is multiplied by 3, so this is a geometric sequence with first term a_1 = 2 and common ratio r = 3.
an=a1⋅r n−1=2⋅3 n−1a_n = a_1 \cdot r^{\,n-1} = 2 \cdot 3^{\,n-1}
Step 2: Substitute n = 10 into the formula.
a10=2⋅3 9a_{10} = 2 \cdot 3^{\,9}
Step 3: Calculate 3 to the 9th power, then multiply by 2.
39=19,683⇒a10=2⋅19,683=39,3663^9 = 19{,}683 \quad \Rightarrow \quad a_{10} = 2 \cdot 19{,}683 = 39{,}366
Answer: The 10th term of the sequence is 39,366.

Frequently Asked Questions

What is the difference between a sequence and a series?
A sequence is a list of numbers in a specific order, while a series is the sum of the terms of a sequence. For example, 2, 4, 6, 8 is a sequence, but 2 + 4 + 6 + 8 = 20 is a series. A sequence lists terms; a series adds them together.
Can a sequence be finite or infinite?
Yes. A finite sequence has a definite number of terms, such as 1, 4, 9, 16 (four terms). An infinite sequence continues without end, indicated by an ellipsis: 1, 4, 9, 16, … The three dots signal that the pattern keeps going indefinitely.

Sequence vs. Series

A sequence is an ordered list of terms, while a series is what you get when you add those terms together. The sequence 1, 2, 3, 4 has four terms listed separately. The corresponding series is 1 + 2 + 3 + 4 = 10. Think of a sequence as the ingredients laid out on a counter, and a series as everything combined in the bowl.

Why It Matters

Sequences appear throughout science, finance, and computer science. Population growth, loan payments, and radioactive decay all follow predictable sequences. Understanding sequences gives you a foundation for studying series, limits, and calculus — the tools used to model change in nearly every quantitative field.

Common Mistakes

Mistake: Confusing the term number n with the term value a_n.
Correction: The term number n is the position in the list (1st, 2nd, 3rd, …), while a_n is the actual value at that position. In the sequence 5, 10, 15, the 2nd term (n = 2) has a value of a_2 = 10, not 2.
Mistake: Assuming every sequence must follow a simple arithmetic or geometric pattern.
Correction: Many sequences follow other rules entirely. The Fibonacci sequence (1, 1, 2, 3, 5, 8, …) is defined by adding the two previous terms. Some sequences are defined by complex formulas or even have no closed-form rule at all.

Related Terms