Types of Sequences and Series: Examples, Formulas & Practice

Types of sequences with examples

These are useful classifications, not an exhaustive list or mutually exclusive categories. A constant nonzero sequence, for instance, is arithmetic with difference 0 and geometric with ratio 1. A short list of terms can also fit many rules; use a stated rule or the intended pattern rather than assuming the continuation is unique.

The familiar harmonic sequence 1, 1/2, 1/3, … is one example. More generally, a harmonic progression has reciprocals in arithmetic progression, as defined by Wolfram MathWorld. The denominators do not have to be consecutive positive integers. They must not be zero where the sequence is defined.

For geometric sequences, a recursive rule “multiply by R” also handles zeros. Ratios involving division by zero cannot be used as a test. Fibonacci-like recurrences with different starting terms are related patterns, but specify their starting values rather than calling every such list the standard Fibonacci sequence.

What is the difference between a sequence and a series?

A sequence lists terms in order. A series is their sum. A finite series stops after a specified number of terms; an infinite series continues without a last term. The Nth partial sum means the sum of the first N terms.

• Sequence: 3, 6, 12, 24, …
• First four terms added: 3 + 6 + 12 + 24 = 45.
• Infinite series: 3 + 6 + 12 + 24 + …, which does not have a finite sum.

The series takes its arithmetic or geometric classification from the sequence being added. “Convergent” and “divergent” describe the behavior of an infinite series, not an extra term pattern to use instead of arithmetic or geometric.

Worked examples: terms and finite sums

In the formulas below, A₁ is the first term, A_N is the Nth term, N is a positive integer, D is the common difference, R is the common ratio, and S_N is the sum of the first N terms.

Arithmetic: find the tenth term and sum

For 2, 5, 8, 11, …, the common difference is D = 3.

1. Use A_N = A₁ + (N − 1)D.
2. The tenth term is A₁₀ = 2 + 9 × 3 = 29.
3. Use S_N = N(A₁ + A_N) ÷ 2.
4. The sum is S₁₀ = 10(2 + 29) ÷ 2 = 155.

Check the sum by pairing the first and last terms: 2 + 29, 5 + 26, and the other three pairs each total 31. Five pairs give 155.

Geometric: find the fifth term and sum

For 3, 6, 12, 24, …, the common ratio is R = 2.

1. Use A_N = A₁R^N−1.
2. The fifth term is A₅ = 3 × 2⁴ = 48.
3. For R ≠ 1, use S_N = A₁(1 − R^N) ÷ (1 − R).
4. The sum is S₅ = 3(1 − 32) ÷ (1 − 2) = 93.

A direct check gives 3 + 6 + 12 + 24 + 48 = 93. If R = 1, use S_N = N × A₁ instead; the fraction formula would divide by zero.

Quadratic: use second differences

For 3, 7, 13, 21, the first differences are 4, 6, 8. Their differences are 2, 2. Continuing that quadratic pattern, the next first difference is 10, so the next term is 21 + 10 = 31. The formula A_N = N² + N + 1 generates these terms for N = 1, 2, 3, ….

When does an infinite series converge?

An infinite series converges when its partial sums approach a finite limit. For an infinite geometric series with nonzero first term, this happens exactly when |R| < 1, and the sum is A₁ ÷ (1 − R). A series of all zeros is a trivial convergent exception.

Example: 4 + 2 + 1 + 1/2 + … has R = 1/2, so its sum is 4 ÷ (1 − 1/2) = 8. The partial sums 4, 6, 7, 7.5, … approach 8. Each finite partial sum is still below 8.

Important: terms approaching zero are necessary for convergence, but are not sufficient. The harmonic series 1 + 1/2 + 1/3 + … diverges even though 1/N approaches zero. Also, a sequence converging and the series of its terms converging are different claims.

For reference, see OpenStax on finite and geometric series formulas and partial sums and infinite-series convergence.

Practice: identify, continue and add

Use the intended standard pattern in each original exercise. Try it before opening the explanation.

1. For 7, 11, 15, 19, …, name the type and find the next term.

   Show answer 1

   Arithmetic: each difference is 4. The next term is 23.

2. For 81, 27, 9, 3, …, find the ratio and next term.

   Show answer 2

   Geometric: R = 1/3. Multiply 3 by 1/3 to get 1.

3. For 2, 6, 12, 20, …, continue the quadratic pattern.

   Show answer 3

   The differences are 4, 6, 8, so the next difference is 10. The next term is 30.

4. Find the sum of the first five terms of 7, 11, 15, 19, ….

   Show answer 4

   The fifth term is 23. The sum is 5(7 + 23) ÷ 2 = 75.

5. Does 6 + 3 + 3/2 + 3/4 + … converge? If so, find its sum.

   Show answer 5

   Yes. The ratio is 1/2, so the sum is 6 ÷ (1 − 1/2) = 12.

Continue with the Arithmetic Sequences, Geometric Sequences, or Recursive and Explicit Definitions of Sequences PDFs in our free Algebra 2 worksheet collection. For matching topic lessons and broader review, visit the Algebra 2 Online Center. Choose the topic that matches your course; these worksheets are not a complete infinite-series calculus course.

How to identify the type of sequence in four checks

1. Subtract neighbors. If every difference is the same, the sequence is arithmetic. Example: 7, 11, 15, 19 has a difference of 4, so the next term is 23.
2. Divide neighbors. If every ratio is the same, it is geometric. Example: 5, 15, 45, 135 has a ratio of 3, so the next term is 405.
3. Take the differences of the differences. If the second differences are constant, it is quadratic. Example: 4, 9, 16, 25 has differences 5, 7, 9 and second differences of 2. These are the square numbers (n + 1)², so the next term is 36.
4. Look for a rule that uses earlier terms. If each term is the sum of the two before it, it is Fibonacci-type. Example: 1, 4, 5, 9, 14 continues with 9 + 14 = 23. If the terms are fractions with 1 on top, flip them and check whether the denominators are arithmetic. If so, it is harmonic.

If none of these work, try comparing each term with its position n. For example, 1, 8, 27, 64 is n³.

Other ways to classify sequences

Besides the pattern type, sequences are often described by how they behave.

• Finite or infinite: 2, 4, 6, 8 stops. 2, 4, 6, 8, … goes on forever.
• Increasing or decreasing: 3, 6, 12 increases. 100, 50, 25 decreases. A constant sequence like 5, 5, 5 does neither.
• Alternating: The signs switch, as in 1, −2, 4, −8 (geometric with ratio −2).
• Convergent or divergent: 1, ½, ¼, ⅛ gets closer and closer to 0, so it converges. 1, 2, 3, 4 grows without limit, so it diverges.
• Explicit or recursive: aₙ = 3n − 1 gives any term directly. aₙ = aₙ₋₁ + 3 builds each term from the one before.

Frequently Asked Questions

What are the types of sequences in math?

The common types of sequences are arithmetic (add the same number each time, like 4, 9, 14, 19), geometric (multiply by the same number, like 2, 6, 18, 54), quadratic (the second differences are constant), harmonic (the reciprocals form an arithmetic sequence), and Fibonacci (each term is the sum of the two before it). Checking differences and ratios usually tells you which one you have.

How many types of sequences are there?

There is no single fixed number. Many textbooks teach four basic types: arithmetic, geometric, harmonic and Fibonacci, and a common list of five adds quadratic sequences. Sequences can also be described in other ways, such as finite or infinite, increasing or decreasing, and convergent or divergent. These labels can overlap, so one sequence may fit more than one description.

What is the difference between sequences and series?

A sequence is an ordered list of numbers, while a series is the sum of those numbers. For example, 1, 4, 7, 10 is a sequence, and 1 + 4 + 7 + 10 = 22 is the matching series. Knowing the sequence type tells you which sum formula to use, such as n(a₁ + aₙ)/2 for an arithmetic series: 4(1 + 10)/2 = 22.

How do you identify the type of sequence?

Start by subtracting consecutive terms. If the difference is constant, the sequence is arithmetic. If not, divide consecutive terms; a constant ratio means geometric. If the first differences change but the second differences are constant, it is quadratic. Finally, check whether each term is the sum of the two before it (Fibonacci) or whether the reciprocals are arithmetic (harmonic).

What are the types of series?

The main types of series match their sequences: arithmetic series, geometric series, harmonic series and power series. A finite geometric series like 3 + 6 + 12 + 24 = 45 can be added with a formula. An infinite geometric series has a finite sum only when the ratio is between −1 and 1, as in 1 + ½ + ¼ + … = 2, while the harmonic series 1 + ½ + ⅓ + … diverges.

Are sequences part of algebra?

Yes. Sequences are usually introduced in Algebra 1, where students write rules for arithmetic and geometric patterns, and they come back in Algebra 2 and Precalculus with series and sigma notation. Infinite series and convergence tests are studied more deeply in calculus. Writing a rule such as aₙ = 3n + 1 is a good bridge between sequences and linear functions.