Recursive Sequence- Definition and Examples

What is a Recursive Sequence?

A recursive sequence is a sequence where each term is defined using the terms that come before it. Instead of giving you a formula to calculate any term directly, a recursive definition tells you how to get from one term to the next.

The classic example: the Fibonacci sequence. You start with two seed values (0 and 1), then each subsequent term is the sum of the two terms before it.

That's it. That's the whole idea. One term depends on the previous terms.

Recursive vs Explicit Formulas

Here's the difference that trips people up:

• Explicit formula: Gives you the nth term directly. You plug in n, you get the answer.
• Recursive formula: Tells you how to get from one term to the next. You need the previous terms first.

Think of it like this: an explicit formula is a shortcut. A recursive formula makes you take the scenic route—you walk through every term to reach your destination.

Common Types of Recursive Sequences

Arithmetic Sequences

Each term increases or decreases by a constant difference. The recursive definition needs a starting value and a common difference.

Example:

a₁ = 5

aₙ₊₁ = aₙ + 3

This gives you: 5, 8, 11, 14, 17...

Geometric Sequences

Each term is multiplied by a constant ratio. Start with a seed value, then keep multiplying.

Example:

a₁ = 2

aₙ₊₁ = aₙ × 3

This gives you: 2, 6, 18, 54, 162...

The Fibonacci Sequence

The most famous recursive sequence. Two seed values, one rule.

a₁ = 1, a₂ = 1

aₙ = aₙ₋₁ + aₙ₋₂

This gives you: 1, 1, 2, 3, 5, 8, 13, 21, 34...

How to Find Terms in a Recursive Sequence

Here's the process, step by step:

1. Identify the base case(s). These are the starting terms given to you directly.
2. Apply the recurrence relation. Use the rule to find the next term.
3. Repeat. Keep applying the rule until you reach the term you need.

Let's work through finding the 6th term of a sequence where a₁ = 4 and aₙ₊₁ = 2aₙ + 1.

a₁ = 4 (given)

a₂ = 2(4) + 1 = 9

a₃ = 2(9) + 1 = 19

a₄ = 2(19) + 1 = 39

a₅ = 2(39) + 1 = 79

a₆ = 2(79) + 1 = 159

The 6th term is 159. No shortcuts—you had to walk through every term.

Solving More Complex Recursive Sequences

Some recursive sequences have patterns that let you find a direct formula. This is called "solving the recurrence."

For linear recurrences like aₙ = aₙ₋₁ + d (arithmetic) or aₙ = r × aₙ₋₁ (geometric), you can derive:

• Arithmetic: aₙ = a₁ + (n-1)d
• Geometric: aₙ = a₁ × rⁿ⁻¹

The Fibonacci sequence is trickier. It has a closed-form solution involving the golden ratio, but that's advanced territory. For most practical purposes, you just iterate.

Where Recursive Sequences Show Up

You see them everywhere once you know what to look for:

• Computer science: Algorithm analysis, recursive functions, fractal generation
• Biology: Population growth models, branching patterns in plants
• Finance: Compound interest calculations, loan amortization
• Mathematics: Fractals, Pascal's triangle, combinatorics

Getting Started: Your First Practice Problems

Problem 1: Find a₅ given a₁ = 3, aₙ₊₁ = aₙ - 2

Answer: 3, 1, -1, -3, -5

Problem 2: Find a₄ given a₁ = 1, a₂ = 4, aₙ = aₙ₋₁ + aₙ₋₂

Answer: 1, 4, 5, 9

Problem 3: Find a₅ given a₁ = 5, aₙ₊₁ = 2aₙ

Answer: 5, 10, 20, 40, 80

Work through these by hand. Write out each term. The only way to get comfortable with recursive sequences is to actually do them.