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Sequences and Series — Definition, Formula & Examples

A sequence is an ordered list of numbers that follow a specific pattern, while a series is the sum of the terms in a sequence. Together, sequences and series form a core topic in precalculus where you analyze patterns, compute sums, and determine whether those sums grow without bound or settle toward a finite value.

A sequence is a function a:N→Ra: \mathbb{N} \to \mathbb{R} that assigns a real number ana_n to each positive integer nn. A series is the formal sum ∑n=1Nan\displaystyle\sum_{n=1}^{N} a_n (finite) or ∑n=1∞an\displaystyle\sum_{n=1}^{\infty} a_n (infinite), where convergence of an infinite series means the sequence of partial sums SN=∑n=1NanS_N = \sum_{n=1}^{N} a_n approaches a finite limit as N→∞N \to \infty.

Key Formula

SN=∑n=1NanS_N = \sum_{n=1}^{N} a_n
Where:
  • ana_n = The nth term of the sequence
  • NN = The number of terms being summed
  • SNS_N = The Nth partial sum (the series value after N terms)

How It Works

To work with a sequence, you first identify its rule — the formula that produces each term from its position number nn. Common types include arithmetic sequences (constant difference between terms) and geometric sequences (constant ratio between terms). Once you have the rule, you can find any term directly. To form a series, you add the terms together; specific summation formulas let you compute finite sums quickly without adding every term one by one. For infinite series, you check whether the partial sums converge to a limit or diverge to infinity.

Worked Example

Problem: Find the sum of the first 5 terms of the arithmetic sequence 3, 7, 11, 15, 19, ...
Identify the pattern: Each term increases by 4, so the common difference is d = 4 and the first term is a₁ = 3.
an=3+(n−1)(4)=4n−1a_n = 3 + (n-1)(4) = 4n - 1
Write the partial sum formula: For an arithmetic series with N terms, use the formula involving the first and last terms.
SN=N2(a1+aN)S_N = \frac{N}{2}(a_1 + a_N)
Substitute values: Here N = 5, a₁ = 3, and a₅ = 19.
S5=52(3+19)=52(22)=55S_5 = \frac{5}{2}(3 + 19) = \frac{5}{2}(22) = 55
Answer: The sum of the first 5 terms is 55.

Another Example

Problem: Determine whether the infinite geometric series 8 + 4 + 2 + 1 + ... converges, and if so, find its sum.
Find the common ratio: Divide any term by the previous term to get r.
r=48=12r = \frac{4}{8} = \frac{1}{2}
Check for convergence: An infinite geometric series converges when |r| < 1. Since |1/2| < 1, this series converges.
Apply the infinite sum formula: Use S = a₁ / (1 − r) for the sum of a convergent geometric series.
S=81−12=812=16S = \frac{8}{1 - \frac{1}{2}} = \frac{8}{\frac{1}{2}} = 16
Answer: The series converges and its sum is 16.

Visualization

Why It Matters

Sequences and series appear throughout AP Precalculus, AP Calculus BC, and college-level analysis courses. In finance, geometric series model compound interest and loan amortization schedules. In engineering and physics, Fourier series — built on this foundation — decompose complex signals into simple wave components.

Common Mistakes

Mistake: Confusing a sequence with a series and writing a list of terms when asked for a sum, or vice versa.
Correction: A sequence is the ordered list of terms; a series is the sum of those terms. Always check whether the problem asks you to list terms or compute a total.
Mistake: Using the infinite geometric sum formula when |r| ≥ 1.
Correction: The formula S = a₁/(1 − r) only works when |r| < 1. If |r| ≥ 1, the series diverges and has no finite sum.

Related Terms