The word “*recursive*” means to recur or repeat. In mathematics, it’s the repeated application of a function or rule to successive results. A **recursive definition of a sequence** defines terms by previous entries in the sequence. It takes the rule for the sequence and applies it over and over, starting with the first term. For example:

- A
_{0}= 0 - A
_{n + 1}= A_{n}+ n + 1

The first term A_{0} 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. A_{0}= 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:

- A
_{0}= 3 - A
_{n + 1}= A_{n}+ 5.

The first few terms of the sequence are:

- A
_{0}=**3**, - A
_{1}= A_{0}+ 5 = 3 + 5 =**8**, - A
_{2}= A_{1}+ 5 = (3 + 5) + 5 =**13**, - A
_{3}= A_{2}+ 5 = [(3 + 5) + 5] + 5 =**18**, - A
_{4}= A_{3}+ 5 = {[(3 + 5) + 5] + 5} + 5 =**23**.

_{1}and f

_{2}= 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:

- A
_{0}=**3**, - A
_{1}= A_{0}+ 5 = 3 + 5, - A
_{2}= A_{1}+ 5 = (3 + 5) + 5, - A
_{3}= A_{2}+ 5 = [(3 + 5) + 5] + 5, - A
_{4}= A_{3}+ 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:

- A
_{0}=**3**, - A
_{1}= A_{0}+ 5 = 3 + 5 - A
_{2}= A_{1}+ 5 = 3 + 2(5), - A
_{3}= A_{2}+ 5 = 3 + 3(5), - A
_{4}= A_{3}+ 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 (A_{3}) 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

**CITE THIS AS:**

**Stephanie Glen**. "Recursive Definition of a Sequence" From

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