A recursively defined sequence is a sequence in which:
- One or more initial terms are specified.
- Each subsequent term is defined in terms of one or more of the preceding terms.
Recursively defined sequences are often used to model situations where the next state depends on the current state, such as population growth, financial investments, or computer algorithms.
Recursive Formulas
A recursive formula for a sequence provides:
- The initial term(s).
- A rule for finding each subsequent term based on the previous term(s).
Example 1: Fibonacci Sequence
The Fibonacci sequence is a classic example of a recursively defined sequence:
- The first two terms are defined as \$F_1 = 1\$ and \$F_2 = 1\$.
- Each subsequent term is the sum of the two preceding terms: \$F_n = F_{n-1} + F_{n-2}\$ for \$n \geq 3\$.
The sequence begins as \$1, 1, 2, 3, 5, 8, 13, \ldots\$
Example 2: Geometric Sequence
A geometric sequence can also be defined recursively:
- The first term is given as \$a_1 = 2\$.
- Each subsequent term is obtained by multiplying the previous term by a common ratio, say \$r = 3\$: \$a_n = 3a_{n-1}\$ for \$n \geq 2\$.
The sequence starts as \$2, 6, 18, 54, \ldots\$
TipWhen working with recursively defined sequences, always identify the initial term(s) and the recursive rule. This will help you generate the sequence correctly.
Generating Terms of a Recursively Defined Sequence
To generate terms of a recursively defined sequence:
- Start with the initial term(s).
- Apply the recursive rule to find the next term.
- Repeat the process to generate as many terms as needed.
Example 3: Recursive Sequence
Consider the sequence defined by:
- \$a_1 = 5\$
- \$a_n = 2a_{n-1} + 1\$ for \$n \geq 2\$
To generate the first five terms:
- \$a_1 = 5\$
- \$a_2 = 2 \cdot 5 + 1 = 11\$
- \$a_3 = 2 \cdot 11 + 1 = 23\$
- \$a_4 = 2 \cdot 23 + 1 = 47\$
- \$a_5 = 2 \cdot 47 + 1 = 95\$
The sequence is \$5, 11, 23, 47, 95, \ldots\$
Theory of KnowledgeHow do recursively defined sequences illustrate the power of simple rules to generate complex patterns? Can you think of other examples in nature or society where recursion plays a role?
Self review1. Define a recursive sequence to model the number of rabbits in a population where each pair of rabbits produces one new pair every month, and rabbits take one month to mature. 2. Generate the first ten terms of the sequence. 3. How does the sequence change if the rabbits take two months to mature?
NoteBe careful not to confuse recursively defined sequences with explicitly defined sequences. In a recursive sequence, each term depends on the previous term(s), while in an explicit sequence, each term is defined independently of the others.
