6.3 Recursive Definitions and Sequences

Recursive definitions and sequences are powerful tools in mathematics. They allow us to create complex patterns from simple rules, starting with initial values and building up step by step. This approach is key in modeling real-world phenomena and solving problems.

In this part, we'll look at famous sequences like Fibonacci, arithmetic, and geometric. We'll also explore functions like factorial. These concepts show how simple rules can create intricate patterns, connecting to the broader ideas of mathematical induction and recursion.

Recursive Definitions

Understanding Recursive Definitions and Components

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• Recursive definition establishes a mathematical object by specifying its initial values and a rule for generating subsequent values
• Initial condition provides the starting point for the recursive process, defining the first term or base case
• Recursive formula determines how to calculate each subsequent term using previous terms in the sequence
• Closed-form expression represents the sequence using a non-recursive formula, allowing direct calculation of any term

Applying Recursive Definitions

• Recursive definitions model real-world phenomena with self-similar patterns (fractals, population growth)
• Initial conditions often represent known values or starting points in a system
• Recursive formulas capture the relationship between consecutive terms, reflecting the pattern of change
• Closed-form expressions enable efficient computation of distant terms without calculating all preceding values

Sequences

Exploring the Fibonacci Sequence

• Fibonacci sequence defined recursively as $F_n = F_{n-1} + F_{n-2}$ with initial conditions $F_0 = 0$ and $F_1 = 1$
• First few terms: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ...
• Appears in nature (spiral patterns in sunflowers, pinecones)
• Golden ratio approximated by the ratio of consecutive Fibonacci numbers as the sequence progresses

Analyzing Arithmetic and Geometric Sequences

• Arithmetic sequence characterized by a constant difference between consecutive terms
  • General term formula: $[a_n](https://www.fiveableKeyTerm:a_n) = [a_1](https://www.fiveableKeyTerm:a_1) + (n - 1)[d](https://www.fiveableKeyTerm:d)$, where $a_1$ is the first term and $d$ is the common difference
  • Sum formula: $[S_n](https://www.fiveableKeyTerm:s_n) = \frac{n}{2}(a_1 + a_n)$, where $n$ is the number of terms
• Geometric sequence defined by a constant ratio between consecutive terms
  • General term formula: $a_n = a_1 * [r](https://www.fiveableKeyTerm:r)^{n-1}$, where $a_1$ is the first term and $r$ is the common ratio
  • Sum formula: $S_n = \frac{a_1(1-r^n)}{1-r}$ for $r \neq 1$, and $S_n = na_1$ for $r = 1$

Functions

Exploring the Factorial Function

• Factorial function denoted as $[n!](https://www.fiveableKeyTerm:n!)$ calculates the product of all positive integers up to and including $n$
• Recursive definition: $n! = n * (n-1)!$ with base case $0! = 1$
• Closed-form expression: $n! = n * (n-1) * (n-2) * ... * 2 * 1$
• Applications include permutations, combinations, and probability calculations
• Grows rapidly, leading to large values even for relatively small inputs (10! = 3,628,800)