Problem 61 - College Algebra with Corequisite Support

Question

Find the tenth term of the sequence \(a_{1}=2, a_{n}=n a_{n-1}\).

Step-by-Step Solution

Answer

The tenth term of the sequence is 7,257,600.

Step 1: Understand the Formula

The sequence is defined recursively, meaning each term is generated based on the previous one. The formula given is \( a_n = n a_{n-1} \), and the first term, \( a_1 \), is 2.

Step 2: Calculate the Second Term

Using the formula \( a_n = n a_{n-1} \) for \( n = 2 \), we find \( a_2 = 2 \times a_1 = 2 \times 2 = 4 \).

Step 3: Calculate the Third Term

Now calculate \( a_3 = 3 \times a_2 = 3 \times 4 = 12 \).

Step 4: Calculate the Fourth Term

Then, find \( a_4 = 4 \times a_3 = 4 \times 12 = 48 \).

Step 5: Calculate the Fifth Term

Next, \( a_5 = 5 \times a_4 = 5 \times 48 = 240 \).

Step 6: Calculate the Sixth Term

Now \( a_6 = 6 \times a_5 = 6 \times 240 = 1440 \).

Step 7: Calculate the Seventh Term

Then \( a_7 = 7 \times a_6 = 7 \times 1440 = 10080 \).

Step 8: Calculate the Eighth Term

Next, \( a_8 = 8 \times a_7 = 8 \times 10080 = 80640 \).

Step 9: Calculate the Ninth Term

Now \( a_9 = 9 \times a_8 = 9 \times 80640 = 725760 \).

Step 10: Calculate the Tenth Term

Finally, \( a_{10} = 10 \times a_9 = 10 \times 725760 = 7257600 \).

Key Concepts

Sequence Terms

Let's dive into understanding sequence terms, as these are the building blocks of any sequence. In mathematics, a sequence is simply an ordered list of numbers. The numbers, known as terms, follow a certain rule to establish their order. In the given exercise, the sequence defined by the recursive relation \( a_{n} = n \cdot a_{n-1} \) starts with the first term \( a_1 = 2 \). This rule tells us precisely how each subsequent term is derived from the one preceding it.

It’s crucial to understand that each term in a sequence has its own unique position, identified by the term’s subscript, \( n \). In this sequence, for example, \( a_2 \) is 4, \( a_3 \) is 12, and so on. Recognizing that sequence terms are linked and dependent on prior terms is key in many areas of mathematics. Each step builds on the previous one.

Factorial Growth

Factorial growth is a fascinating mathematical concept that describes how certain sequences increase rapidly. In our example of the sequence \( a_{n} = n \cdot a_{n-1} \), each term is the product of \( n \) and the previous term. This growth pattern resembles that of a factorial structure, where \( n! = n \times (n-1) \times (n-2) \times \ldots \times 2 \times 1 \).

As you calculate terms further into the sequence, you’ll notice the values increasing dramatically. Each term multiplies more factors together, resulting in numbers that grow increasingly large. This type of growth is non-linear and happens quickly; it helps to understand how factorial operations work to appreciate the behavior of such sequences.

• Factorials feature in permutations and combinations.
• Used in calculations of exponential growth problems.

Recursive Formula

A recursive formula is a fundamental concept in mathematics, particularly when dealing with sequences. In essence, it’s an equation that defines each term of a sequence depending on the preceding terms. The exercise's recursive formula \( a_n = n \times a_{n-1} \) is a great example. The formula requires an initial value, or base case, to kick off the calculations; in this situation, it’s given as \( a_1 = 2 \).

The recursive nature means each new term is calculated using operations on the previous term, ensuring that without the base case, the progression of the sequence would be undefined. Recursive formulas provide a compact way to express sequences, often simplifying problem-solving approaches by reducing the need for explicit list relationships.

• Ideal for computing complex sequences efficiently.
• Can efficiently solve iterative problems.

Problem Solving in Algebra

Algebra problem-solving often involves breaking down complex tasks into smaller, more manageable parts. The recursive nature of the sequence in the exercise demonstrates a clear approach. Starting from a known base value and working through step-by-step calculations to solve, you reach subsequent terms or solutions.

This method not only builds understanding but also helps in finding solutions accurately and efficiently. Practicing recursive sequences in algebra is a powerful way to improve problem-solving skills.

Here are a few tips to enhance your algebra problem-solving skills:

• Always begin by understanding the given problem.
• Identify the base case and understand its role.
• Step through each part of the sequence methodically.
• Keep calculations organized to avoid errors.

These strategies will provide a blueprint to approaching many algebra problems involving sequences.