Recursive vs. Explicit Rules

There are two ways to describe the same sequence. An explicit rule is a formula in terms of $n$ — plug in the term number and out comes the term, like $a_n = 5 + 3(n - 1)$ . A recursive rule instead tells you where to start and how to step: $a_1 = 5,\ a_n = a_{n-1} + 3$ .

Both rules generate the identical list of numbers. The skill is recognizing which kind you are looking at, converting between them, and knowing which one is faster for the question in front of you.

How to tell them apart

Look at what the rule depends on. If the formula uses $n$ — the term's position — it is explicit. If it uses $a_{n-1}$ — the previous term — it is recursive.

A complete recursive rule always has two parts: a starting value like $a_1 = 5$ , and a step like $a_n = a_{n-1} + 3$ . Without the starting value, the step rule alone cannot pin down the sequence — you would know how to move but not where to begin.

Converting between the two

For an arithmetic sequence, both rules are built from the same two numbers: the first term $a_1$ and the common difference $d$ . The explicit rule is $a_n = a_1 + d(n - 1)$ ; the recursive rule is $a_1 = \text{(first term)},\ a_n = a_{n-1} + d$ .

So converting is just reading off $a_1$ and $d$ and rewriting. From $a_n = 3 + 8(n - 1)$ , you read $a_1 = 3$ and $d = 8$ , and the recursive version is $a_1 = 3,\ a_n = a_{n-1} + 8$ .

Which rule is faster?

For a nearby term, either works — stepping three times is easy. For a far-away term like $a_{50}$ , the explicit rule wins: it jumps straight there in one substitution, while the recursive rule needs $49$ separate steps.

Recursive rules shine when the sequence is naturally described step by step — a balance that changes each month, a pattern that grows each stage. Explicit rules shine when you need any single term on demand.

Worked examples

Example 1: classify a rule

Is $a_n = 5 + 3(n - 1)$ explicit or recursive?

Check what the formula depends on

a_n = 5 + 3(n - 1)

It uses

n

, not the previous term

a_{n-1}

A formula in terms of

n

is explicit

Answer: Explicit

Example 2: use a recursive rule

A sequence is defined by $a_1 = 7$ and $a_n = a_{n-1} + 3$ . Find $a_3$ .

Start at the first term

a_1 = 7

Add

3

for the second term

a_2 = 10

Add

3

again

a_3 = 13

Answer: $a_3 = 13$

Example 3: write both rules

Write the explicit and recursive rules for $4, 10, 16, 22, \ldots$

Read the first term

a_1 = 4

Find the common difference

d = 10 - 4 = 6

Explicit rule from

a_n = a_1 + d(n - 1)

a_n = 4 + 6(n - 1)

Recursive rule: start and step

a_1 = 4,\ a_n = a_{n-1} + 6

Answer: Explicit: $a_n = 4 + 6(n - 1)$ ; recursive: $a_1 = 4,\ a_n = a_{n-1} + 6$

Try one yourself

Common questions

What does

a_{n-1}

mean?

It is the term right before $a_n$ — the previous term. A rule like $a_n = a_{n-1} + 6$ says: to get any term, take the one before it and add $6$ .

Why does a recursive rule need a first term?

The step rule only tells you how to move from one term to the next. Without a starting value there is nothing to step from, so the actual numbers stay unknown. A complete recursive rule always states $a_1$ .

Do geometric sequences have both kinds of rules too?

Yes. A geometric sequence with first term $a_1$ and ratio $r$ has explicit rule $a_n = a_1 \cdot r^{n-1}$ and recursive rule $a_1 = \text{(first term)},\ a_n = a_{n-1} \cdot r$ . Multiplying replaces adding.

Which rule should I use to find

a_{50}

The explicit rule. It jumps straight to any term in one substitution. The recursive rule would make you compute all $49$ terms before it.

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