Question

Describe the pattern in each sequence, and use the pattern to find the next three terms. $$12,6,3, \frac{3}{2}, \frac{3}{4}, \frac{3}{8}, \dots$$

Short answer

The next three terms are \( \frac{3}{16}, \frac{3}{32}, \frac{3}{64} \).

Step-by-step solution

Identify the pattern

Observe the given sequence: \(12, 6, 3, \frac{3}{2}, \frac{3}{4}, \frac{3}{8}\). Notice that each term is obtained by multiplying the previous term by \( \frac{1}{2} \).

Find the next term

To find the next term, multiply \( \frac{3}{8} \) by \( \frac{1}{2} \): \[ \frac{3}{8} \times \frac{1}{2} = \frac{3}{16} \].

Find the second term

Multiply \( \frac{3}{16} \) by \( \frac{1}{2} \): \[ \frac{3}{16} \times \frac{1}{2} = \frac{3}{32} \].

Find the third term

Multiply \( \frac{3}{32} \) by \( \frac{1}{2} \): \[ \frac{3}{32} \times \frac{1}{2} = \frac{3}{64} \].

Key concepts

Patterns in Sequences

Patterns in sequences are essential for predicting future terms. In arithmetic and geometric sequences, you’ll find fixed rules or operations that link consecutive numbers. For this sequence: \(12, 6, 3, \frac{3}{2}, \frac{3}{4}, \frac{3}{8}\), notice that each term is generated by multiplying the previous one by \( \frac{1}{2} \). Identifying this recurring change allows us to extend the sequence accurately. Recognizing these patterns is useful in various fields like coding, science, and even finance.

Multiplication of Fractions

When dealing with sequences involving fractions, it's crucial to understand fraction multiplication. This concept helps you handle patterns in more complex sequences. To multiply fractions, follow this simple rule: multiply the numerators (top numbers) together, and the denominators (bottom numbers) together. For example, when calculating the next term of our sequence: \( \frac{3}{8} \times \frac{1}{2} \), you multiply \(3 \times 1\) to get \(3\) and \(8 \times 2\) to get \(16\), thus resulting in \( \frac{3}{16} \). This same process applies as you progress through subsequent terms.

Geometric Progression

Geometric progression is a common type of sequence where each term is a constant multiple of the previous one. This constant factor is known as the common ratio. In our sequence, the common ratio is \( \frac{1}{2} \). Starting from the first term, you multiply by this ratio to get the next term: \(12 \times \frac{1}{2} = 6\), \(6 \times \frac{1}{2} = 3\), and so forth. Understanding geometric progression helps in many applications, such as calculating interest rates, growth patterns, and even in natural phenomena. Recognizing the ratio and applying it becomes straightforward once you grasp the concept.