Math · Precalculus · Concept

Arithmetic and geometric sequences

A sequence is an ordered list of numbers, given by an explicit formula for the nth term or by a recursive formula that builds each term from the one before. An arithmetic sequence adds a common difference d at each step, and a geometric sequence multiplies by a common ratio r. Each kind has formulas for its nth term and for the sum of its first n terms.

Terms and formulas

aₙ is the term in position n. An explicit formula gives aₙ directly from n; a recursive formula gives each term from the one before it, starting from a given first term.

a_{n} = 3 n + 2 or a_{1} = 5, a_{n + 1} = a_{n} + 3

\begin{array}{c} a_{n} = 3 n + 2 \\ or a_{1} = 5, a_{n + 1} = a_{n} + 3 \end{array}

Arithmetic sequences

Each term is the one before plus the common difference d. Reaching term n from a₁ takes n − 1 steps of d, so the terms lie on a line: aₙ is a linear function of n with slope d.

a_{n} = a_{1} + (n - 1) d

Geometric sequences

Each term is the one before times the common ratio r. Reaching term n takes n − 1 multiplications, so the terms follow an exponential function of n. A ratio between −1 and 1 makes the terms shrink toward 0, and a negative ratio makes their signs alternate.

a_{n} = a_{1} r^{n - 1}

Telling them apart

Subtract consecutive terms: a constant difference means arithmetic. Divide them: a constant ratio means geometric. Many sequences are neither, such as the squares 1, 4, 9, 16.

The sum of an arithmetic sequence

Pair the first term with the last, the second with the next-to-last, and so on: every pair has the same total. So the sum of n terms is n times the average of the first and last terms.

S_{n} = \frac{n (a_{1} + a_{n})}{2}

The sum of a geometric sequence

Multiply the sum by r and subtract it from the original: every term but two cancels. The formula needs r ≠ 1; when r = 1 every term is a₁ and the sum is na₁.

S_{n} = \frac{a_{1} (1 - r^{n})}{1 - r}, r \neq 1

Sigma notation

Σ writes a sum in one symbol: the index runs from the lower limit to the upper limit, and each value gives one term. Adding infinitely many terms of a geometric sequence gives a finite total only when |r| < 1, which is where infinite series begin.

\sum_{k = 1}^{20} (3 k + 2) = 670

Common mistakes

Using n instead of n − 1 in the nth-term formula: a₁ is reached after zero steps.
Taking the ratio upside down: for 48, 24, 12, … the ratio is 24/48 = 1/2, not 2.
Using the arithmetic sum formula for a geometric sequence, or the other way round.
Miscounting the terms: from 5 to 62 in steps of 3 there are (62 − 5)/3 + 1 = 20 terms, not 19.

Key terms

Sequence: An ordered list of values, with each term identified by an integer index. A sequence may be given by a direct formula or a rule based on earlier terms.
Arithmetic sequence: A sequence in which each term is the one before plus a fixed common difference d, so aₙ = a₁ + (n − 1)d. The sum of its first n terms is n(a₁ + aₙ)/2.
Geometric sequence: A sequence in which each term is the one before times a fixed common ratio r, so aₙ = a₁rⁿ⁻¹. For r ≠ 1 the sum of its first n terms is a₁(1 − rⁿ)/(1 − r).
Partial sum: The sum of the first n terms of a series, Sₙ = a₁ + ⋯ + aₙ. If the partial sums approach a limit, that limit is the series’ sum.
Sigma notation: A compact way to write a sum: Σ from i = 1 to n of aᵢ means a₁ + a₂ + ⋯ + aₙ. The index i takes each integer value from the lower limit to the upper limit, and each value contributes one term.
Geometric series: A series in which each term is the previous one times a fixed ratio r: a + ar + ar² + ⋯. It converges to a/(1 − r) when |r| < 1 and diverges otherwise, unless a = 0.

Work through an example

For the arithmetic sequence 5, 8, 11, 14, …, find the 20th term and the sum of the first 20 terms.

Find a term and a sum of an arithmetic sequence →

Find a term and a sum of a geometric sequence →

Decide if a sequence is arithmetic or geometric →

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List the terms in Sequences & infinite series Check the term and the sum in Math Open worked example on a board Arithmetic sum in Math Reference

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