Problem 5 - Prealgebra

Question

Find the next term in list. \(3,12,48,192,768, \dots\)

Step-by-Step Solution

Answer

The next term is 3072.

Step 1: Identify the Pattern

Examine the given sequence: 3, 12, 48, 192, 768. Notice that each term appears to be a multiple of the previous term. Let's identify this pattern. Divide each term by its preceding term: \( \frac{12}{3} = 4, \frac{48}{12} = 4, \frac{192}{48} = 4 \), and \( \frac{768}{192} = 4 \). Each term is the previous term multiplied by 4.

Step 2: Calculate the Next Term

Now that we have the pattern, we can calculate the next term in the sequence. Since each term is the previous term times 4, take the last term, 768, and multiply it by 4: \( 768 \times 4 = 3072 \).

Key Concepts

Understanding Sequence Patterns

When you first look at a sequence, it might seem like just a row of numbers. However, there's often a pattern underlying these numbers. Identifying this pattern is crucial to predicting what comes next.
In the exercise, we have this sequence: 3, 12, 48, 192, 768, ... . At a glance, it might not be obvious what the pattern is. By examining how one number relates to the next, we can uncover it. In this case, each number is obtained by multiplying the previous number by 4. This type of consistent relationship from one term to the next is what we refer to as the sequence pattern.

• Understanding these patterns helps in predicting future values.
• It provides a systematic way to handle and solve sequence problems.

The beauty of recognizing sequence patterns is that once identified, you can continue the series indefinitely, as long as the rule stays the same.

Prealgebra Concepts in Sequences

Before diving into more complex math, prealgebra helps to build a strong foundation in understanding numbers and their relationships. Sequences, like the one given in the exercise, are a great way to practice this.
Prealgebra concepts bring together:

• Identifying relationships between numbers, like addition, subtraction, multiplication, and division.
• Recognizing patterns and formulating rules based on these patterns.
• Applying these rules systematically to solve problems.

In our sequence, understanding prealgebra helps identify that we multiply by 4 to transition from one term to the next. Such exercises enhance your ability to recognize numerical patterns intuitively, laying the groundwork for more advanced algebraic thinking.

Multiplication in Sequences

In sequences, multiplication can serve as the core operation that connects each number to the next. This method of interaction often forms what is known as a "geometric sequence".
Here's a step by step understanding of its application:

• Start with the first term in the sequence.
• Multiply by a fixed number (in this case, 4) to determine the next term.
• Continue this process to generate additional terms.

For the given sequence, each term is generated by multiplying the previous term by 4. Thus:

• 3 × 4 = 12
• 12 × 4 = 48
• 48 × 4 = 192
• 192 × 4 = 768
• And, 768 × 4 = 3072, the next term.

This multiplication process is both systematic and consistent, which is a hallmark of geometric sequences. By understanding this, you're more equipped to tackle similar sequence problems in the future.