Problem 96 - Prealgebra

Question

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

Step-by-Step Solution

Answer

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

Step 1: Identify the Pattern

To identify the pattern in this geometric sequence, observe the ratio between successive terms. Divide each term by its preceding term to find this common ratio.

Step 2: Calculate the Common Ratio

Calculate the common ratio: \[\text{Common ratio} = \frac{1}{\frac{2}{3}} = \frac{3}{2}\]\[\frac{3}{2} \div 1 = \frac{3}{2}\]\[\frac{9}{4} \div \frac{3}{2} = \frac{3}{2}\]The common ratio is \( \frac{3}{2} \) for the sequence.

Step 3: Apply the Common Ratio

To find the next number, multiply the last given term by the common ratio \( \frac{3}{2} \). So, the next number is: \[\frac{9}{4} \times \frac{3}{2} = \frac{27}{8}\]Thus, the next number in the sequence is \( \frac{27}{8} \).

Key Concepts

Common Ratio

In a geometric sequence, the magic number is the *common ratio*. This ratio remains consistent and ties each term to the one before it. The common ratio is the crucial clue that reveals the growth or decay pattern of the sequence. To find this, pick any term in the sequence (starting from the second one) and divide it by the term immediately before it. For the sequence given, the common ratio can be found by:

• Dividing 1 by \(\frac{2}{3}\) gives \(\frac{3}{2}\).
• Checking \(\frac{3}{2}\) divided by 1 also results in \(\frac{3}{2}\).
• And \(\frac{9}{4}\) divided by \(\frac{3}{2}\) again provides the same value \(\frac{3}{2}\).

Each time, the result is \(\frac{3}{2}\), which confirms our common ratio. This constant value is essential because it defines the character of the sequence.

Sequence Pattern

The concept of sequence pattern is all about understanding how each term in the sequence relates to others. Once we have the common ratio, we can map out the whole sequence effortlessly. In a geometric sequence, this relationship is multiplicative. Begin with the first term, and use the common ratio to find subsequent terms by multiplying:

• The first term is \(\frac{2}{3}\).
• The second term becomes \((\frac{2}{3}) \times (\frac{3}{2}) = 1\).
• The process repeats for later terms.

Every term differs from its predecessor by multiplication with \(\frac{3}{2}\). This knowledge helps when predicting what comes next.

Term Calculation

Calculating the next term in a geometric sequence is akin to following a recipe. With the ingredients (each preceding term) and a repeated method (the common ratio), finding any term becomes simple.To calculate the next term in our given sequence:* Start with the last known term: \(\frac{9}{4}\).* Multiply it by the common ratio: \(\frac{9}{4} \times \frac{3}{2}\).* Calculating gives \(\frac{27}{8}\), the next term in the sequence.By applying this method, we’re drawing on the power of repeated multiplication to reveal the hidden pattern within the sequence, guiding us to the correct term.