Problem 46 - Algebra 2

Question

Find the next four terms of each arithmetic sequence. \(\frac{18}{5}, \frac{16}{5}, \frac{14}{5}, \ldots\)

Step-by-Step Solution

Answer

The next four terms are \(\frac{12}{5}, 2, \frac{8}{5}, \frac{6}{5}\).

Step 1: Identify the Common Difference

In an arithmetic sequence, the difference between consecutive terms is constant and is called the common difference. Subtract the second term from the first term to find it: \( \frac{16}{5} - \frac{18}{5} = -\frac{2}{5} \). The common difference is \(-\frac{2}{5}\).

Step 2: Determine the Next Term

To find the next term of the sequence, add the common difference to the last given term (\( \frac{14}{5} \)): \( \frac{14}{5} + \left(-\frac{2}{5}\right) = \frac{12}{5} \). The next term is \( \frac{12}{5} \).

Step 3: Calculate the Fourth Term

Use the common difference to determine the term after \( \frac{12}{5} \) by adding it to \( \frac{12}{5} \): \( \frac{12}{5} + \left(-\frac{2}{5}\right) = \frac{10}{5} = 2 \). The subsequent term is 2.

Step 4: Calculate the Fifth Term

Continue to add the common difference to find the term after 2: \( 2 + \left(-\frac{2}{5}\right) = \frac{10}{5} - \frac{2}{5} = \frac{8}{5} \). The next term is \( \frac{8}{5} \).

Step 5: Calculate the Sixth Term

Finally, add the common difference to \( \frac{8}{5} \): \( \frac{8}{5} + \left(-\frac{2}{5}\right) = \frac{6}{5} \). The last term of the sequence is \( \frac{6}{5} \).

Key Concepts

Common Difference

In arithmetic sequences, one of the most vital concepts to grasp is the common difference. This is the consistent value we add (or subtract) to each term to find the next one. For instance, in the given sequence, \( \frac{18}{5} \), \( \frac{16}{5} \), \( \frac{14}{5} \), the common difference is calculated by subtracting the first term from the second. Here, it results in \( \frac{16}{5} - \frac{18}{5} = -\frac{2}{5} \). This negative value indicates that each subsequent term is smaller by \( \frac{2}{5} \) compared to the previous one.

• Consistent throughout the sequence.
• Determines the rate of change between terms.

Consecutive Terms

Consecutive terms in a sequence are terms that appear one after the other. In arithmetic sequences, we build one term from the previous one using the common difference. For example, starting from \( \frac{14}{5} \), adding the common difference \( -\frac{2}{5} \) gives us \( \frac{12}{5} \), which appears directly after \( \frac{14}{5} \).Understanding this concept:

• Ensures accurate calculations.
• Keeps the sequence aligned with the expected pattern.

The flow of calculation continues similarly for other terms in the sequence.

Sequence Pattern

The pattern in an arithmetic sequence is fundamental for determining future terms. Each term follows the sequence rule established by the starting number and the common difference.

• Starts with an initial value.
• Progresses steadily due to the common difference.
• Predictable and structured.

For the current sequence, each new term is computed by subtracting \( \frac{2}{5} \) from the previous term, establishing a reliable and repeatable pattern. Recognizing this pattern helps in projecting the next terms without calculations.

Fractions in Sequences

Sequences can involve fractions just as easily as whole numbers. Handling fractions might seem tricky but follows the same arithmetic principles. In our sequence, all terms are fractions with a denominator of 5. This uniformity simplifies adding or subtracting the common difference.When working with fractions:

• Ensure denominators are the same for easy addition or subtraction.
• Apply arithmetic operations as you would with regular numbers.

For example, calculating between terms such as \( \frac{12}{5} \) and \( \frac{10}{5} \), keep the denominator unchanged for straightforward manipulation. Such sequences can improve fluency in both arithmetic operations and fractional understanding.