Recursive Formula

Recursive Formula

For a sequence a₁, a₂, a₃, . . . , a_n, . . . a recursive formula is a formula that requires the computation of all previous terms in order to find the value of a_n .

Note: Recursion is an example of an iterative procedure.

Example of recursive formula: a₁=3, aₙ=2aₙ₋₁+5, showing steps a₁=3 through a₅=123.

See also

Explicit formula

Key Formula

a_n = f(a_{n-1}), \quad \text{with a given } a_1

Where:

$a_n$ = The nth term of the sequence
$a_{n-1}$ = The previous term in the sequence
$a_1$ = The first term (the initial condition that starts the sequence)
$f$ = A rule that relates each term to the preceding term(s)

Worked Example

Problem: A sequence is defined by the recursive formula a₁ = 3, aₙ = aₙ₋₁ + 5. Find the first five terms.

Step 1: Start with the given initial term.

a_1 = 3

Step 2: Apply the recursive rule to find the second term.

a_2 = a_1 + 5 = 3 + 5 = 8

Step 3: Use the second term to find the third term.

a_3 = a_2 + 5 = 8 + 5 = 13

Step 4: Continue the pattern to find the fourth and fifth terms.

a_4 = a_3 + 5 = 13 + 5 = 18

Step 5: Compute the fifth term.

a_5 = a_4 + 5 = 18 + 5 = 23

Answer: The first five terms are 3, 8, 13, 18, 23.

Another Example

This example uses a geometric (multiplicative) recursive rule rather than the arithmetic (additive) rule in the first example, showing that recursive formulas can define many types of sequences.

Problem: A sequence is defined by a₁ = 2, aₙ = 3·aₙ₋₁. Find a₅.

Step 1: Write down the initial term.

a_1 = 2

Step 2: Multiply by 3 to get each successive term. Find a₂.

a_2 = 3 \cdot a_1 = 3 \cdot 2 = 6

Step 3: Find a₃ and a₄.

a_3 = 3 \cdot 6 = 18, \quad a_4 = 3 \cdot 18 = 54

Step 4: Find the fifth term.

a_5 = 3 \cdot 54 = 162

Answer: a₅ = 162.

Frequently Asked Questions

What is the difference between a recursive formula and an explicit formula?

A recursive formula gives the next term based on previous terms (e.g., aₙ = aₙ₋₁ + 5), so you must know prior terms to compute a later one. An explicit formula gives any term directly as a function of n (e.g., aₙ = 5n − 2), letting you jump straight to any term without computing the ones before it.

Why do recursive formulas need an initial condition?

The recursive rule only describes how to get from one term to the next. Without a starting value like a₁ = 3, you would have no anchor, and infinitely many different sequences could satisfy the same rule. The initial condition pins down a unique sequence.

When should you use a recursive formula instead of an explicit formula?

Recursive formulas are natural when a problem describes how each step depends on the previous one — for instance, population growth, compound interest, or the Fibonacci sequence. If you need to find a term far along in the sequence (like the 1000th term) quickly, converting to an explicit formula is usually more efficient.

Recursive Formula vs. Explicit Formula

	Recursive Formula	Explicit Formula
Definition	Defines aₙ in terms of previous term(s)	Defines aₙ directly as a function of n
Example (arithmetic)	a₁ = 3, aₙ = aₙ₋₁ + 5	aₙ = 5n − 2
Finding the 100th term	Must compute all 99 preceding terms first	Substitute n = 100 directly
Initial condition needed?	Yes — always required	No — built into the formula
Best for	Problems where each step depends on the last	Quickly finding any specific term

Why It Matters

Recursive formulas appear throughout algebra and precalculus courses whenever sequences and series are studied. They also form the basis for understanding algorithms in computer science, where recursion is a fundamental programming technique. Real-world models — such as compound interest, population dynamics, and the famous Fibonacci sequence — are most naturally expressed recursively.

Common Mistakes

Mistake: Forgetting to state the initial condition (e.g., writing only aₙ = aₙ₋₁ + 5 without specifying a₁).

Correction: A recursive formula is incomplete without an initial value. Always include something like a₁ = 3 so the sequence is uniquely determined.

Mistake: Trying to find a distant term (like a₅₀) by plugging n = 50 directly into the recursive rule as if it were an explicit formula.

Correction: A recursive formula requires you to compute every term in order. If you need a distant term quickly, first convert the recursive formula to its explicit form.

Related Terms

Sequence — The ordered list a recursive formula generates
Explicit Formula of a Sequence — The alternative way to define a sequence directly
Iterative Process — Recursion is an example of iteration
Formula — General concept that recursive formulas are a type of
Arithmetic Sequence — Common sequence type often defined recursively
Geometric Sequence — Another common type with a multiplicative recursive rule
Term — Each individual element produced by the formula
Compute — Each term must be computed from previous ones