Recursive Definition of a Sequence

Sequence and series >

A recursive definition of a sequence defines terms by previous entries in the sequence. For example:

• A₀ = 0
• A_{n + 1} = A_n + n + 1

The first term A₀ is defined explicitly, as zero. This is the starting point— you need this to build a recursive definition. Every subsequent term in the sequence is defined recursively in terms of A_n. In other words, you need two parts for a recursive definition of a sequence:

• The initial term (e.g. A₀ = 0),
• A recurrence relation (also called a recursion formula): a symbolic description of subsequent terms (e.g. A_{n + 1} = A_n + n + 1).

The other way to define sequences is with a general term, also called an explicit definition. For example, the sequence defined by a_n = 1/n only requires you to plug in a value for n. You do not need to know the initial term in order to solve the sequence.

Recursive Definition of a Sequence: Examples

The following sequence is defined recursively with an initial term and a rule for subsequent terms:

• A₀ = 3
• A_{n + 1} = A_n + 5.

The first few terms of the sequence are:

• A₀ = 3,
• A₁ = A₀ + 5 = 3 + 5 = 8,
• A₂ = A₁ + 5 = (3 + 5) + 5 = 13,
• A₃ = A₂ + 5 = [(3 + 5) + 5] + 5 = 18,
• A₄ = A₃ + 5 = {[(3 + 5) + 5] + 5} + 5 = 23.

The Fibonacci sequence is defined recursively with two initial conditions: f₁ and f₂ = 1 and the recursion formula f_n = f_n-1 + f_n-2 for n ≥ 3. Each term in the sequence (after the first two) is the sum of the previous two terms. The first few terms are: 1, 1, 3, 5, 8, 13, 21,….

How to Find the Explicit Definition from a Recursive Definition of a Sequence

There isn’t a formula you can follow to turn a recursive definition to an explicit definition of a sequence. However, there are a few general steps that you can use to identify the pattern needed for an explicit definition.

Step 1: Write out the first few terms of the sequence using the recursive definition, without actually carrying out the arithmetic operations. Using the example above:

• A₀ = 3,
• A₁ = A₀ + 5 = 3 + 5,
• A₂ = A₁ + 5 = (3 + 5) + 5,
• A₃ = A₂ + 5 = [(3 + 5) + 5] + 5,
• A₄ = A₃ + 5 = {[(3 + 5) + 5] + 5} + 5.

Step 2: Look for ways to combine terms. A quick glance at the above terms tells us that 5 is repeating, so we can combine them to get:

• A₀ = 3,
• A₁ = A₀ + 5 = 3 + 5
• A₂ = A₁ + 5 = 3 + 2(5),
• A₃ = A₂ + 5 = 3 + 3(5),
• A₄ = A₃ + 5 = 3 + 4(5).

From here, we can see a pattern emerge: each step in the process multiplies 5 by the term’s index (the index is the subscript in A_n). For example, the third (A₃) has 3 * 5. That leads to the general formula:
a_n = 3 + 5n.

References

Lameda, B. & Nikolaev, N. (2016). Integral Calculus. Retrieved February 4, 2021 from: http://www.math.toronto.edu/beatriz/files/MAT137/MAT136_Lecture_Notes.pdf

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