Problem 56 - Precalculus with Limits

Question

In Exercises 47-62, write an expression for the apparent \(n\)th term of the sequence. (Assume that \( n \) begins with 1.) \( 1, \dfrac{1}{2}, \dfrac{1}{6}, \dfrac{1}{24}, \dfrac{1}{120}, \dots \)

Step-by-Step Solution

Answer

The apparent nth term of the sequence is given by the expression \(1/n!\).

Step 1: Identify the pattern

Look at the sequence and try to identify a pattern. In this case, one can observe that the denominator in each fraction increases by multiplying by the next natural number. This pattern corresponds to the factorial function, which can be written as \(n!\), where \(n!\) is the product of all positive integers less than or equal to \(n\).

Step 2: Formulate the apparent nth term

With the pattern identified in step 1, it's reasonable to suggest that each term in the sequence corresponds to \(1/n!\), where \(n\) is the term number. The proposed nth term would thus be \(1/n!\).

Step 3: Verify the hypothesis

Now let's check the hypothesis against the given sequence to verify its validity. If the hypothesis is correct, the \(1/n!\) expression should produce each term in the sequence when \(n\) is substituted with the term number.\nFor the first term, where \(n = 1\), \(1/1! = 1\).For the second term, where \(n = 2\), \(1/2! = 1/2\).For the third term, where \(n = 3\), \(1/3! = 1/6\).For the fourth term, where \(n = 4\), \(1/4! = 1/24\).For the fifth term, where \(n = 5\), \(1/5! = 1/120\).As these results match the sequence exactly, the hypothesis that the nth term is \(1/n!\) is verified.

Key Concepts

Apparent nth Term

Understanding the concept of an apparent nth term is crucial when dealing with sequences in mathematics. The apparent nth term represents a general formula that can generate any term in a sequence based on its position. It is the 'n' in the nth term that denotes this position. Here's a closer look at the apparent nth term in the context of our exercise:

The sequence begins with 1, and subsequent terms are created by dividing 1 by the factorial of the term number. Since factorials grow quickly, each term in the sequence gets smaller. To identify the apparent nth term, we utilized the pattern recognized and formulated that for any term at position 'n', the value would be the inverse of the factorial of 'n'. Therefore, the apparent nth term for our sequence is \( \frac{1}{n!} \). This elegant expression encompasses the entire sequence reliably and accurately.

The process of finding the apparent nth term often involves trial and error, pattern recognition, and verification. For students, mastering the technique of expressing the nth term is invaluable for solving sequence problems, especially in more advanced mathematics, and provides a clear path to calculating any term within a sequence.

Factorial Function

The factorial function, symbolized by an exclamation point \( n! \), is a fundamental concept in mathematics, particularly in combinatorics and sequence analysis. A factorial represents the product of all positive integers up to a given number. In essence:

\[ n! = n \times (n - 1) \times (n - 2) \times \dots \times 2 \times 1 \]
For example, \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \). It's also important to note that by convention, \( 0! = 1 \).

This function grows very rapidly with each incremental increase of 'n'. In our sequence problem, the denominator of each fraction consists of the factorial of the term number. Understanding the factorial function is crucial to solving many problems in mathematics and is a building block for various topics such as permutations, combinations, probability, and series. When faced with a sequence like ours, recognizing the involvement of the factorial function can dramatically simplify the process of identifying the pattern and thus determining the apparent nth term of the sequence.

Sequence Pattern Recognition

Sequence pattern recognition is a critical thinking skill that involves identifying the consistent changes or operations that define the progression of a sequence. It is essential for students to practice and develop this skill to recognize patterns quickly and accurately in mathematical problems.

In our exercise, the ability to observe that each term in the given sequence was derived through the factorial function represents a successful application of sequence pattern recognition. By recognizing the pattern, one can deduce not only the apparent nth term but also anticipate future terms. This skill is particularly useful in advanced areas of study such as calculus, where understanding the behavior of a series can help in determining convergence or divergence.

To hone this skill, one must look for common relationships between terms, such as addition, subtraction, multiplication, division, or more complex operations like factorials or powers. Through regular practice, the application of sequence pattern recognition can be extended beyond textbook exercises and utilized in analyzing real-world phenomena, where patterns often emerge in data and can inform predictions and decision-making.