Problem 34 - Contemporary Precalculus

Question

Find the first five terms of the recursively defined sequence. $$a_{0}=1, a_{1}=1, \text { and } a_{n}=n a_{n-1} \quad \text { for } n \geq 2$$

Step-by-Step Solution

Answer

Answer: The first five terms of the sequence are 1, 1, 2, 6, and 24.

Step 1: Identify the first two terms

The first two terms are given: \(a_0=1\) and \(a_1=1\).

Step 2: Find the third term (n=2)

Use the provided formula \(a_n = na_{n-1}\), when \(n = 2\): $$a_2 = 2a_{2-1} = 2a_1 = 2(1) = 2$$

Step 3: Find the fourth term (n=3)

Use the provided formula, when \(n = 3\): $$a_3 = 3a_{3-1} = 3a_2 = 3(2) = 6$$

Step 4: Find the fifth term (n=4)

Use the provided formula, when \(n = 4\): $$a_4 = 4a_{4-1} = 4a_3 = 4(6) = 24$$

Step 5: Find the sixth term (n=5)

Use the provided formula, when \(n = 5\): $$a_5 = 5a_{5-1} = 5a_4 = 5(24) = 120$$

Step 6: Write the first five terms of the sequence

The first five terms of the sequence are: $$a_0 = 1, a_1 = 1, a_2 = 2, a_3 = 6, a_4 = 24$$

Key Concepts

Factorial Sequence

A factorial sequence is a specific type of sequence where each term is defined based on the factorial of a number.
The factorial of a number (denoted as \(n!\)) is the product of all positive integers up to that number.
For example, \(4!\) is calculated as \(4 \times 3 \times 2 \times 1 = 24\).
In the context of the sequence from the exercise, we see that the sequence \(a_n\) is defined so that each term after the first two is constructed by multiplying the current index \(n\) by the previous term \(a_{n-1}\).
This effectively leads to the calculation of factorials in a recursive manner.

• \(a_0 = 1\) and \(a_1 = 1\) serve as the base of the sequence.
• \(a_2 = 2 \times a_1 = 2 \times 1 = 2\) (which is \(2!\))
• \(a_3 = 3 \times a_2 = 3 \times 2 = 6\) (which is \(3!\))
• \(a_4 = 4 \times a_3 = 4 \times 6 = 24\) (which is \(4!\))

This pattern continues for each subsequent term, effectively making this a factorial sequence.

Sequence Terms

A sequence is essentially an ordered list of numbers following some rule, where each number is a term.
In the exercise, the terms of the sequence are generated starting from an initial condition or base.
For the factorial sequence discussed:

• The sequence starts with two predefined terms, \(a_0 = 1\) and \(a_1 = 1\).


• These base terms are crucial as they serve as the foundation for generating the rest of the sequence.


• Subsequent terms rely on the previous ones, building onto them to create a complex pattern.

Every term in this particular sequence is calculated using a defined recursive formula, meaning each term is derived from the previous term, creating a dependency chain from the first term to the last. The way each term builds on the one before makes sequences like these both predictable and intriguing.

Recursion Formula

The recursive formula is a powerful tool in mathematics used to define a sequence in terms of prior terms.
In other words, it's a way to express a complex problem with simpler versions of itself. For the sequence provided, the recursion formula is given by:

• \(a_n = n \times a_{n-1}\), applicable for \(n \geq 2\).


• This formula means each term is the product of its index and the preceding term in the sequence.

Using this formula helps in constructing each subsequent term starting from the base cases \(a_0\) and \(a_1\).
This formula makes it easy to extend the sequence further without manually computing each term from scratch.
By breaking down the task repetitively as each term is calculated from the previous, it showcases the elegance and utility of recursion in mathematical sequence building.