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 or . An initial condition, usually , 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 and for .
- In an arithmetic sequence, each term is found by adding a constant difference: .
- The explicit formula for an arithmetic sequence is .
- In a geometric sequence, each term is found by multiplying by a constant ratio: .
- The explicit formula for a geometric sequence is .
- The common difference is found by subtracting consecutive terms: .
- The common ratio is found by dividing consecutive nonzero terms: .
- A recursive rule is best for building terms step by step, while an explicit formula is best for finding a term like 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 for the term in position .
- 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 .
- 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 does not say where to start.
- Using instead of in an explicit formula is wrong when the first term is , 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 .
- Calling a sequence geometric without checking ratios is wrong because geometric sequences must have a constant ratio .
- Trying to find recursively when an explicit formula is available is inefficient because recursion requires calculating every earlier term first.
Practice Questions
- 1 Given and , find , , , and .
- 2 Write a recursive rule and an explicit formula for the arithmetic sequence .
- 3 For the geometric sequence , find and write the recursive rule.
- 4 A pattern is described by and . Explain why this is recursive and why it is not a simple geometric sequence.
Understanding Recursive Sequences & Patterns
A useful first step is to inspect how a list changes rather than how large the numbers look. Subtract each term from the next one. If the results stay fixed, the pattern has a constant additive change.
If they do not, try dividing consecutive terms when division makes sense. A fixed multiplier points to repeated scaling.
This distinction matters because adding the same amount creates steady straight line growth, while multiplying by the same amount creates growth or decay that speeds up over time. A sequence can begin with small values yet become huge when it is repeatedly multiplied.
The position number is part of every sequence rule. Students often make an off by one error when matching a term to its position. Write a short table with one column for the term number and one column for the value.
Check that the first value matches position one before using a formula. For example, a savings plan that begins at twenty dollars and gains five dollars each week has twenty dollars at week one, not week zero, unless the situation clearly defines the starting moment differently.
Words in a problem decide the index. A deposit made before any weeks pass may be called term zero.
Recursive thinking appears in many settings where the current state comes from the previous state. A staircase pattern may add a fixed number of tiles in each new figure. A bouncing ball may reach a fixed fraction of its earlier height after every bounce.
Population models may multiply by a growth factor, although real populations usually stop following one simple rule when food, space, or disease become important. Loan balances are often recursive too.
Interest increases the balance, then a payment reduces it. Such a rule may involve more than one operation, so it is not purely arithmetic or geometric.
Not every recursive sequence has a simple explicit formula. Some rules need two earlier terms. In the Fibonacci pattern, each new value is found by adding the two values before it.
That means two starting values are required. Other rules can produce irregular behavior even when the instructions are short. When working with these patterns, calculate several terms carefully and keep the order clear.
Negative differences make arithmetic sequences decrease. Negative ratios make geometric sequences switch signs.
A ratio of zero makes every later term zero. Checking a proposed rule against at least three transitions helps catch errors before they spread through a table or graph.