A sequence can be described in two fundamentally different ways. A recursive rule tells you how to get the next term from earlier terms. An explicit rule tells you how to jump directly to any chosen position.
Both descriptions can be correct for the same sequence, but they support different kinds of reasoning. This guide compares them, shows how to translate between them in familiar cases, and explains why recognising the difference becomes important in algebra, functions and later mathematics.
Recursive description
For the sequence 5, 8, 11, 14, … a recursive description is:
T1 = 5
Tn = T(n-1) + 3, for n ≥ 2This says: start at 5, then add 3 to the previous term.
Explicit description
The same sequence has explicit rule:
Tn = 3n + 2This lets us find T100 immediately without calculating T99 first.
What each rule is good at
| Recursive rule | Explicit rule |
|---|---|
| Shows how the sequence evolves | Gives direct access to any position |
| Natural for processes and iteration | Natural for algebra and distant terms |
| Needs earlier terms | Usually uses n directly |
| Can model dependence on previous states | Can reveal position-to-value structure |
Worked example 1: arithmetic sequence
Sequence: 12, 17, 22, 27, …
Recursive form:
T1=12
Tn=T(n-1)+5Explicit form:
Tn=12+5(n-1)=5n+7The two rules describe the same sequence from different viewpoints.
Worked example 2: geometric growth
Sequence: 3, 6, 12, 24, …
Recursive form:
T1=3
Tn=2T(n-1)Explicit form:
Tn=3×2^(n-1)The recursive rule makes the repeated doubling obvious. The explicit rule makes long jumps easy.
Worked example 3: a sequence that depends on more than one previous term
Consider 1, 1, 2, 3, 5, 8, … Each new term is the sum of the previous two.
T1=1, T2=1
Tn=T(n-1)+T(n-2)The recursive description is natural because the process itself depends on earlier states. Not every useful sequence is best introduced through a simple explicit school-level formula.
Initial conditions are part of a recursive rule
The instruction “add 4 each time” does not define one unique sequence. Starting at 1 gives 1,5,9,… while starting at 10 gives 10,14,18,…
A recursive rule therefore needs enough starting information to determine the sequence.
Recursive thinking models processes
Bank balances, populations, repeated depreciation and iterative algorithms can all be described recursively because the next state depends on the current state.
For example, a quantity that grows by 5% each period can be modelled recursively as:
Tn = 1.05T(n-1)with an initial value specified separately.
Explicit thinking supports direct comparison
When an explicit rule is available, comparing two sequences can become an algebra problem. Instead of generating many terms, solve where their formulas are equal or compare their growth directly.
The same sequence can have many-looking descriptions
Tn=4n-1 and Tn=3+4(n-1) look different, but they are algebraically equivalent. Likewise, a recursive rule can encode the same arithmetic structure through repeated addition.
Students should compare meaning, not surface appearance.
Diagnostic table
| Observed mistake | Likely issue | Repair |
|---|---|---|
| Writes “+3” as complete recursive rule | Missing initial condition | State the first term |
| Uses recursive rule to find T100 term-by-term | Representation choice | Convert to explicit form when possible |
| Thinks explicit rule shows process better | Purpose confusion | Compare evolution versus direct position |
| Confuses T(n-1) with Tn-1 | Notation parsing | Read T(n-1) as “previous term” |
A tutor’s teaching route
- Generate a sequence from a verbal repeated rule.
- Write the initial condition.
- Formalise the recursive notation.
- For arithmetic sequences, derive the explicit nth term.
- Compare what each representation makes easy.
- Introduce a multi-step recurrence to show why recursion can be the natural form.
Practice
- Write a recursive rule for 7, 12, 17, 22, …
- Write an explicit rule for the same sequence.
- Write a recursive rule for 5, 15, 45, 135, …
- Why does a recursive rule usually need an initial condition?
Answers
1. T1=7, Tn=T(n-1)+5. 2. Tn=5n+2. 3. T1=5, Tn=3T(n-1). 4. Because the recurrence tells how to continue, but the starting value determines which sequence is generated.
Connected routes
Use From Pattern Spotting to a General Term, Arithmetic Sequences, and Quadratic Patterns as companion guides. Return to the Mathematics Learning Hub.