Problem 8 - Algebra 2

Question

Describe each pattern formed. Find the next three terms. $$ 1,2,6,24,120, \dots $$

Step-by-Step Solution

Answer

720, 5040, 40320

Step 1: Identify the Pattern

First, look at the sequence and try to identify any apparent pattern. It seems that each number in the sequence is the product of its index (starting from 1) and the previous number in the sequence.

Step 2: Confirm the Pattern

Is this pattern consistent? For example, the third term is \(6 = 3 \times 2\), the fourth term is \(24 = 4 \times 6\), and the fifth term is \(120 = 5 \times 24\). Our observation from the first step can thus be confirmed.

Step 3: Use the Pattern

Now, use the pattern to determine the next three terms by simply multiplying the latest term by the next index. The sixth term would be \(720 = 6 \times 120\), the seventh term would be \(5040 = 7 \times 720\) and the eighth term would be \(40320 = 8 \times 5040\).

Step 4: Display the Full Sequence

Finally, display the complete sequence, including the three computed terms: 1, 2, 6, 24, 120, 720, 5040, 40320.

Key Concepts

Mathematical Patterns

Mathematical patterns are sequences that follow a specific rule or set of rules. Recognizing these patterns is crucial in understanding and solving many mathematical problems. In our given sequence, the pattern involves factorials. You can notice that each term is obtained by multiplying the previous term by its position in the sequence. This is a common technique used to generate what we call `factorials`. A factorial, represented by an exclamation mark (!), means you multiply a series of descending natural numbers. For instance, in this sequence:

• The first number is 1.
• The second number is 1 x 2 = 2, which is the factorial of 2.
• The third number is 2 x 3 = 6, which is the factorial of 3.

Recognizing this pattern allows you to predict future terms effectively. A mastery of mathematical patterns empowers you to understand the logic behind sequences and can lead to success in more complex math problems.

Sequence Analysis

Sequence analysis is the process of examining the components of a sequence to understand its structure and predict future elements. With our current sequence, the key is to notice that each term multiplies with an increasing integer. If you break down the sequence:

• Start with the number 1.
• Multiply by 2 to get the next term, 2.
• Continue by multiplying by 3, yielding 6, and so on.

This consistent pattern shows us that the structure is defined by multiplying integers sequentially. In sequence analysis, it's helpful to first observe, then verify the pattern by checking multiple terms, as was done in our original exercise. By confirming the pattern, you can confidently forecast the next terms in the sequence. Doing so not only solves this specific problem but sharpens the analytical skills needed for different types of mathematical sequences.

Algebraic Thinking

Algebraic thinking involves using variables and algebraic expressions to understand and describe patterns and relationships. In the context of our sequence, algebraic thinking helps us establish a generalized formula for the sequence terms using the concept of factorials. The key here is realizing that each term can be expressed as: \( n! \), where \( n \) represents the position of the term.
For example:

• The second term is \( 2! = 2 \)
• The third term is \( 3! = 6 \)
• The fourth term is \( 4! = 24 \)

By recognizing this factorial expression, one ties the sequence to a well-known algebraic concept, which simplifies understanding and extends this understanding to a broader class of problems. This approach allows you to create functions and algorithms that can model similar sequences efficiently. Engaging in algebraic thinking like this is a fundamental skill for moving from simple arithmetic to more advanced mathematics.