Question

Tell how to find the next term in each pattern. $$3,6,12,24, \ldots$$

Short answer

The next term is 48.

Step-by-step solution

Identify the Pattern

Observe the sequence: 3, 6, 12, 24. Notice that each term is being multiplied by 2 to get the next term. This suggests a pattern of multiplication.

Confirm the Pattern

Check the multiplication rule for each pair of consecutive terms: - 6 = 3 x 2 - 12 = 6 x 2 - 24 = 12 x 2. The pattern is consistent. Each term is indeed the previous term multiplied by 2.

Apply the Pattern to Find the Next Term

To find the next term, take the last term given, which is 24, and multiply by 2. Thus, the next term is 24 x 2 = 48.

Key concepts

Pattern Recognition

Recognizing patterns is an essential skill for solving sequence problems in mathematics. The first step involves observation, where you carefully examine the given sequence to identify any regularities or trends. For the sequence 3, 6, 12, 24, it's clear that each number is stepping up in size. By investigating further, we notice that each term can be reached by multiplying the previous term by 2.

• Look for relationships: Consider if there is a consistent mathematical operation between the terms like addition, subtraction, multiplication, or division.
• Check for consistency: Once you suspect a pattern, verify it across multiple terms to ensure it's applied consistently.
• Make predictions: After recognizing and confirming the pattern, use it to predict the next terms in the sequence.

Geometric Sequences

A geometric sequence is a sequence where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the "common ratio." In the sequence 3, 6, 12, 24, the common ratio is 2. This means each term is double the one before it.
To analyze geometric sequences:

• Identify the first term: Here, the first term "a" is 3.
• Find the common ratio: The common ratio "r" is determined by dividing any term by the preceding term. For example, \( \frac{6}{3} = 2 \).
• Use the formula: The nth term of a geometric sequence can be expressed as \( a imes r^{(n-1)} \).

This structured approach allows you to compute any term in the sequence. Understanding geometric sequences helps you recognize how exponential patterns are formed.

Multiplication Patterns

Multiplication patterns are evident in sequences like our example—where each number is derived by multiplying the previous one by a constant. Recognizing this can simplify your understanding of number series. These patterns are foundational elements in many areas of math. Observing such sequences helps in predicting future values.
To work with multiplication patterns:

• Detect the pattern: Inspect how each term relates to the previous one. Is a specific multiplication factor consistently applied?
• Verify the pattern: Check that the suspected pattern holds true over several term transitions. For instance, multiplying 3 by 2 yields 6, and this sequence of multiplication continues.
• Project the pattern: To find further terms, continue applying the multiplication. For example, for the sequence 3, 6, 12, 24, multiplying 24 by 2 gets 48.

These steps create a way to both understand and extend sequences efficiently. Recognizing multiplication patterns broadens your toolkit for tackling a vast array of mathematical problems.