Generate sequence terms from rules

Algebra • Sequences

Flashcards

Test your knowledge with interactive flashcards

Given the sequence 5, 8, 11, 14..., state the term-to-term rule.

Click to reveal answer

Add 3 to get the next term.

Key concepts

What you'll likely be quizzed about

Definition of a sequence

A sequence is an ordered list of numbers called terms. Each term occupies a position labelled by a positive integer, often written as n. The first term corresponds to n = 1 unless another starting index is stated.

Term-to-term rule

A term-to-term rule gives an operation that transforms one term into the next. Examples include add 3, multiply by 2, or subtract 5. Repeating the operation produces subsequent terms, so a single rule governs all adjacent pairs of terms.

Position-to-term (nth term) rule

A position-to-term rule assigns a formula for the nth term as a function of n, for example 3n + 1. Substituting the position number n into the formula generates the corresponding term directly. The formula removes the need to calculate earlier terms to reach later ones.

Arithmetic (constant difference) sequences

A sequence with the same difference between consecutive terms follows a term-to-term rule 'add d' and a position-to-term rule an + b where a equals the common difference. Constant difference causes linear growth or decline, so the nth term is linear in n.

Recognising the appropriate rule

First differences reveal the type of rule: constant first differences indicate an arithmetic (linear) position-to-term rule. Non-constant differences require testing for multiplication rules or other patterns. The chosen rule type determines whether the sequence is generated by repeated operations or by a direct formula.

Limiting factors and domain

Positions (n) require positive integers or a stated alternative domain. Some rules produce non-integer or negative terms, which remain valid unless the context restricts values. Term-to-term rules require a known starting term; position-to-term rules require a correct formula for all intended n values.

Key notes

Important points to keep in mind

Term-to-term rules require a known starting term.

Position-to-term rules allow direct computation of any term by substituting n.

Check first differences to detect arithmetic (linear) sequences.

Confirm the domain of n; positions are usually positive integers.

Test a proposed nth term by substituting small n values.

Arithmetic sequences have nth term of the form an + b where a is the common difference.

Multiplicative term-to-term rules produce geometric sequences unless modified by addition.

Ensure clarity whether sequences start at n = 0 or n = 1 before using a position-to-term rule.