Recursive Sequences & Patterns Cheat Sheet

Recursive sequences describe patterns by giving a starting value and a rule for finding each next term. Students need this cheat sheet to recognize patterns, write recursive rules, and connect recursion to tables and formulas. It is especially useful for arithmetic and geometric sequences, where the same type of change repeats from one term to the next.

The core idea is that a recursive rule depends on earlier terms, such as $a_n = a_{n-1} + d$ or $a_n = r \cdot a_{n-1}$ . An initial condition, usually $a_1$ , tells where the sequence begins. Arithmetic sequences add a constant difference, while geometric sequences multiply by a constant ratio.

Many problems ask students to move between recursive rules, explicit formulas, tables, and written pattern descriptions.

Key Facts

A recursive definition needs an initial value and a rule, such as $a_1 = 5$ and $a_n = a_{n-1} + 3$ for $n \ge 2$ .
In an arithmetic sequence, each term is found by adding a constant difference: $a_n = a_{n-1} + d$ .
The explicit formula for an arithmetic sequence is $a_n = a_1 + (n - 1)d$ .
In a geometric sequence, each term is found by multiplying by a constant ratio: $a_n = r \cdot a_{n-1}$ .
The explicit formula for a geometric sequence is $a_n = a_1 \cdot r^{n-1}$ .
The common difference is found by subtracting consecutive terms: $d = a_n - a_{n-1}$ .
The common ratio is found by dividing consecutive nonzero terms: $r = \frac{a_n}{a_{n-1}}$ .
A recursive rule is best for building terms step by step, while an explicit formula is best for finding a term like $a_{50}$ directly.

Vocabulary

Sequence: A sequence is an ordered list of numbers that usually follows a pattern or rule.
Term: A term is one number in a sequence, often written as $a_n$ for the term in position $n$ .
Recursive Rule: A recursive rule defines each new term using one or more previous terms.
Initial Condition: An initial condition is the starting value of a recursive sequence, such as $a_1 = 4$ .
Common Difference: The common difference is the constant amount added to get from one term to the next in an arithmetic sequence.
Common Ratio: The common ratio is the constant factor multiplied to get from one term to the next in a geometric sequence.

Common Mistakes to Avoid

Forgetting the initial value makes the recursive definition incomplete because the rule $a_n = a_{n-1} + d$ does not say where to start.
Using $n$ instead of $n - 1$ in an explicit formula is wrong when the first term is $a_1$ , because the first term has zero repeated changes.
Calling a sequence arithmetic just because it increases is wrong because arithmetic sequences must have a constant difference $d$ .
Calling a sequence geometric without checking ratios is wrong because geometric sequences must have a constant ratio $r = \frac{a_n}{a_{n-1}}$ .
Trying to find $a_{50}$ recursively when an explicit formula is available is inefficient because recursion requires calculating every earlier term first.

Practice Questions

1 Given $a_1 = 7$ and $a_n = a_{n-1} + 4$ , find $a_2$ , $a_3$ , $a_4$ , and $a_5$ .
2 Write a recursive rule and an explicit formula for the arithmetic sequence $3, 8, 13, 18, \ldots$ .
3 For the geometric sequence $2, 6, 18, 54, \ldots$ , find $r$ and write the recursive rule.
4 A pattern is described by $a_1 = 10$ and $a_n = 2a_{n-1} - 3$ . Explain why this is recursive and why it is not a simple geometric sequence.

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Recursive Sequences & Patterns Cheat Sheet

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