Problem 47 - Algebra 2

Question

Find the next four terms of each arithmetic sequence. \(6.7,6.3,5.9, \ldots\)

Step-by-Step Solution

Answer

The next four terms are 5.5, 5.1, 4.7, and 4.3.

Step 1: Identify the Common Difference

To find the common difference in an arithmetic sequence, subtract the first term from the second term. Here, the second term is 6.3 and the first is 6.7.\( d = 6.3 - 6.7 = -0.4 \)So, the common difference is -0.4.

Step 2: Calculate the 5th Term

Use the last known term and the common difference to calculate the next term. The 4th term is found by subtracting the common difference from the last given term, which is 5.9.\( 5.9 - 0.4 = 5.5 \)Therefore, the 5th term is 5.5.

Step 3: Calculate the 6th Term

Use the 5th term to find the 6th term by again subtracting the common difference.\( 5.5 - 0.4 = 5.1 \)The 6th term is 5.1.

Step 4: Calculate the 7th Term

Repeat the process using the 6th term.\( 5.1 - 0.4 = 4.7 \)The 7th term is 4.7.

Step 5: Calculate the 8th Term

Finally, subtract the common difference from the 7th term to find the 8th term.\( 4.7 - 0.4 = 4.3 \)The 8th term is 4.3.

Key Concepts

Common Difference

In an arithmetic sequence, the common difference is a key element. It is the number that you add (or subtract) consistently to each term to get the next one. Knowing the common difference is crucial because it governs the entire pattern of the sequence.
For example, let's examine the sequence: 6.7, 6.3, 5.9, ...

• Calculate the common difference by subtracting the first term from the second: \( d = 6.3 - 6.7 = -0.4 \)
• A negative common difference means each term is smaller than the one before it.

This makes it straightforward to predict future terms in the sequence. Without the common difference, we wouldn't know how the sequence grows or shrinks.

Sequence Terms

Sequence terms are the individual numbers that make up an arithmetic sequence. They follow a specific order based on a repeated pattern dictated by the common difference. In our example:

• The sequence starts with 6.7, then continues to 6.3, 5.9, and so forth.
• Each term is derived by shifting the previous term by the common difference of -0.4.

This regular pattern is integral as it ensures the sequence maintains its unbroken continuity. Understanding how each term relates to the others through the common difference lays the groundwork for predicting future terms efficiently.

Arithmetic Progression

An arithmetic progression is another term for an arithmetic sequence. It's a series of numbers where the difference between consecutive terms remains constant. This type of progression offers a straightforward way to deal with numeric sequences.
In real world examples:

• Arithmetic progressions can project population growth or decay, financial interest spread, or even steps in computer algorithms.
• The mathematical structure supports various calculations to determine extensive patterns using a known formula: \( a_n = a_1 + (n-1) \times d \), where \( a_n \) is the n-th term, \( a_1 \) is the first term, and \( d \) is the common difference.

These calculations help in extending the known sequence by predicting further terms without needing to calculate each in sequence.

Term Calculation

Term calculation in an arithmetic sequence involves using the common difference to find unknown terms. You start with known terms and add or subtract the common difference until reaching the desired term.
For the sequence starting with 6.7, the term calculation uses repetitive subtraction:

• 4th term is 5.9. Subtract 0.4 to find 5th term: \( 5.9 - 0.4 = 5.5 \).
• Continue this process: \( 5.5 - 0.4 = 5.1 \) (6th term), and \( 5.1 - 0.4 = 4.7 \) (7th term).
• So, predicting terms is about applying the common difference continually to derive future terms.

Such calculations form a foundational component of understanding arithmetic sequences and are broadly applicable in various mathematical scenarios.