Problem 15 - Algebra 2

Question

Find the next two terms of each geometric sequence. $$ 81,108,144, \dots $$

Step-by-Step Solution

Answer

The next two terms are 192 and 256.

Step 1: Understanding the Problem

You have a geometric sequence: 81, 108, 144, ... You need to determine the next two terms in this sequence by identifying the common ratio.

Step 2: Finding the Common Ratio

In a geometric sequence, each term after the first can be found by multiplying the previous term by the common ratio (\(r\)). To find \(r\), divide the second term (108) by the first term (81): \[r = \frac{108}{81} = \frac{4}{3}\]

Step 3: Calculating the Fourth Term

Use the common ratio \(r = \frac{4}{3}\) to find the next term (fourth term) by multiplying the third term (144) by \(r\): \[144 \times \frac{4}{3} = 192\] Thus, the fourth term is 192.

Step 4: Calculating the Fifth Term

Again, apply the common ratio to the newly found fourth term (192) to find the fifth term: \[192 \times \frac{4}{3} = 256\] Thus, the fifth term is 256.

Key Concepts

Understanding the Common Ratio

In any geometric sequence, the common ratio is a key component. It is the constant factor that you multiply by each term to get to the next one. To find the common ratio, you divide a term in the sequence by the one that precedes it. For example, given the sequence 81, 108, 144,... we find the common ratio by dividing 108 by 81. This gives us:

• \( r = \frac{108}{81} = \frac{4}{3} \)

Once you have the common ratio, you can use it to predict future terms in the sequence. Each multiplication takes you one step further along, making the sequence grow. Remember to apply the common ratio consistently to ensure the sequence remains geometric. It's like having a map to navigate through the sequence terms!

Identifying Sequence Terms

In a geometric sequence, the sequence terms are the numbers in the specific order we're interested in. Starting with a known term, we use the common ratio to derive the next. Each term is evenly spaced from the last, not by addition, but through multiplication.
Consider the sequence 81, 108, 144,... where each term grows larger by multiplying by the common ratio \( r = \frac{4}{3} \). For instance, to find the next term after 144, we multiply:

• Fourth term: \( 144 \times \frac{4}{3} = 192 \)

And again, to find what comes after this new term, we'll multiply once more:

• Fifth term: \( 192 \times \frac{4}{3} = 256 \)

This process of using the common ratio keeps the sequence orderly, and predictable.

Performing Mathematical Calculations

Mathematical calculations in geometric sequences are essential for deducting new terms based on the common ratio. These calculations often involve straightforward multiplication and division, enhancing our fluency with numbers.
When working through the sequence 81, 108, 144,... with a common ratio of \( \frac{4}{3} \), here's what happens behind the scenes:

• To find a term, determine the product of the previous term and the common ratio. For the third term (144), the next calculation would be:
  \[ 144 \times \frac{4}{3} = 192 \]
• Repeat this calculation to move to subsequent terms:
  \[ 192 \times \frac{4}{3} = 256 \]

Each step is methodical, ensuring consistency in deriving new terms. These calculations illustrate how applying a mathematical rule can expand a series consistently, building on what you know to uncover what comes next.