Problem 37 - Calculus Single Variable

Question

Give a recursive definition of the sequence. $$ f_{n}=n !, n=1,2,3, \ldots $$

Step-by-Step Solution

Answer

Sequence is defined recursively as \( f_1 = 1 \) and \( f_n = n \times f_{n-1} \) for \( n > 1 \).

Step 1: Define the Base Case

The base case provides a simple, non-recursive answer for the smallest input for which the sequence is defined. For the factorial sequence, we set the base case as follows:\[ f_1 = 1! = 1 \]

Step 2: Define the Recursive Step

The recursive step defines how to compute the term \( f_{n+1} \) using the previously defined terms. For the factorial sequence, each term \( f_{n} \) is defined as \( n \times f_{n-1} \). Thus, the recursive step is:\[ f_{n} = n \times f_{n-1}, \text{ for } n > 1 \]

Step 3: Complete the Recursive Definition

Combine the base case and the recursive step to provide a complete recursive definition for the sequence:1. Base case: \( f_1 = 1 \).2. Recursive step: \( f_{n} = n \times f_{n-1} \) for \( n > 1 \).

Key Concepts

Factorial Sequence

The factorial sequence is a fundamental concept in mathematics where each term in the sequence is the product of a number and all the positive integers below it. This sequence is often used in mathematics for permutations, combinations, and other probability calculations. The factorial of a non-negative integer, expressed as "n!", is defined as:

• \( n! = n \times (n-1) \times (n-2) \times \,...\, \times 2 \times 1 \)
• For example, \( 4! = 4 \times 3 \times 2 \times 1 = 24 \)

In a recursive context, this sequence can be defined with a base case and a recursive step, allowing it to be expressed in terms of its previous elements. Understanding this sequence through recursion helps in implementing efficient mathematical algorithms and solving complex problems in computer science and combinatorics.

Base Case

The base case in a recursive definition is critical as it ensures that the recursion terminates. It's the simplest instance of the problem and provides a straightforward answer without further recursion. In the factorial sequence, the base case is defined as:

• \( f_1 = 1! = 1 \)

This indicates that the factorial of 1 is simply 1. The base case acts as an anchor for the recursive sequence, guaranteeing that the sequence does not recurse indefinitely. By establishing a base case, we create a foundation that every recursive problem must eventually reach, ensuring that recursive computations are both defined and finite.

Recursive Step

The recursive step is the mechanism by which larger problems are broken down into smaller, more manageable pieces. It defines the relationship between the terms of the sequence, connecting each term to its predecessor. For the factorial sequence, the recursive step is expressed as:

• \( f_{n} = n \times f_{n-1} \), for \( n > 1 \)

This step explains that any factorial for a number 'n' is calculated by multiplying 'n' with the factorial of its predecessor, \( f_{n-1} \). This recursive relationship allows us to compute factorials efficiently by reusing already computed values. The careful construction of the recursive step makes it possible to solve for any term in the sequence by following a repeatable and systematic process.

Mathematical Induction

Mathematical induction is a proof technique used to verify properties of sequences and equations, especially when recursion is involved. It relies on proving that if a statement holds for one case, and if it holds for an arbitrary case, it must hold for the next case too. This can effectively demonstrate the validity of recursively defined sequences.To use induction in the context of the factorial sequence:1. **Base Case:** Verify the property for the base case \( n = 1 \). - \( f_1 = 1 \,\Rightarrow \, \text{True for the base case.} \)2. **Inductive Step:** Assume it holds for an arbitrary integer \( n = k \), i.e., \( f_k = k! \). - Now show it holds for \( n = k+1 \): \[ f_{k+1} = (k+1) \times f_k = (k+1) \times k! = (k+1)! \]By confirming both the base case and the inductive step, we can conclude through mathematical induction that the entire sequence is valid for all natural numbers. This technique is crucial to validate the correctness of recursive sequences and ensure they work as expected across all instances.