Problem 95 - Prealgebra

Question

Find the next number in each of the geometric sequences below. \(\frac{3}{2}, 1, \frac{2}{3}, \frac{4}{9}, \ldots\)

Step-by-Step Solution

Answer

The next number in the sequence is \( \frac{8}{27} \).

Step 1: Identify the Common Ratio

In a geometric sequence, each term is obtained by multiplying the previous term by a constant, called the common ratio. To find the common ratio, divide any term by the previous term. For example, divide the second term by the first term: \( \frac{1}{\frac{3}{2}} = \frac{2}{3} \). This means the common ratio \(r\) is \(\frac{2}{3}\).

Step 2: Verify the Common Ratio

To ensure the sequence has a consistent common ratio, check other pairs of consecutive terms: \( \frac{\frac{2}{3}}{1} = \frac{2}{3}\) and \( \frac{\frac{4}{9}}{\frac{2}{3}} = \frac{2}{3} \). Each division confirms that the common ratio is \( \frac{2}{3} \).

Step 3: Calculate the Next Term

Use the common ratio to find the next term in the sequence. Multiply the last known term, \( \frac{4}{9} \), by the common ratio \( \frac{2}{3} \): \[ \frac{4}{9} \times \frac{2}{3} = \frac{8}{27} \]. Thus, the next term is \( \frac{8}{27} \).

Key Concepts

Common Ratio in Geometric Sequences

In a geometric sequence, understanding the "common ratio" is crucial. The common ratio is a constant factor that you multiply by to get from one term to the next. If you know any two consecutive terms, you can find this ratio by dividing the second term by the first. For example, consider the sequence: \(\frac{3}{2}, 1, \frac{2}{3}, \frac{4}{9}\). Here, you divide 1 by \(\frac{3}{2}\) to determine the common ratio as \(\frac{2}{3}\). You can double-check your work by confirming that this ratio applies between other consecutive terms. This consistent factor ensures that the pattern is maintained throughout the sequence. Confident in your common ratio, you can predict any term in the sequence by repeated multiplication of this factor.

Understanding Fractions in Sequences

Fractions may seem tricky, but they are just numbers expressed in terms of parts of a whole. In geometric sequences, fractions often appear because they beautifully illustrate ratios and growth or decay in steps smaller than whole numbers. For instance, in our sequence, each term is a fraction: \(\frac{3}{2}, 1, \frac{2}{3}, \frac{4}{9}\).Operations with fractions are straightforward once you remember a few basic rules:

• To multiply fractions, multiply the numerators and multiply the denominators.
• When dividing fractions, multiply by the reciprocal of the divisor.

Fractions help efficiently define patterns in sequences, showing the subtle changes in terms rather than jumping by whole numbers. This is why understanding fractions is important when working with geometric sequences.

Identifying Sequence Patterns

Recognizing patterns in sequences is like solving a puzzle. In geometric sequences, the pattern involves a consistent multiplication process, defined by the common ratio. Each term is derived by multiplying the previous one by this "magic" number. Let's see how easily you can find these patterns.In our sequence \(\frac{3}{2}, 1, \frac{2}{3}, \frac{4}{9}\), observe that every term is achieved by multiplying the term before it by \(\frac{2}{3}\). Start with \(\frac{3}{2}\), multiply by \(\frac{2}{3}\), and you get 1. Multiply 1 by \(\frac{2}{3}\) and reach \(\frac{2}{3}\), and so forth.Once you've identified the pattern, it's like uncovering how the sequence "works." It gives you the power to calculate further terms without starting from the beginning, just by using multiplication repeatedly. This systematic approach not only solves the problem but gives you insight into predictable behavior within geometric structures.