Problem 7 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 3-10: Find the \(n\)th term, the fifth term, and the tenth term of the arithmetic sequence. $$ -7,-3.9,-0.8,2.3, \ldots $$

Step-by-Step Solution

Answer

The nth term is \(3.1n - 10.1\). The 5th term is \(5.4\), and the 10th term is \(20.9\).

Step 1: Identify the common difference

In an arithmetic sequence, the difference between each pair of consecutive terms (known as the common difference \(d\)) is constant. The first two terms in this sequence are \(-7\) and \(-3.9\). Calculate \(d\) by subtracting the first term from the second term: \((-3.9) - (-7) = 3.1\). Hence, \(d = 3.1\).

Step 2: Write the general formula for the n-th term

The formula for the \(n\)th term \(a_n\) in an arithmetic sequence is given by \(a_n = a_1 + (n-1) \, d\), where \(a_1\) is the first term and \(d\) is the common difference. For this sequence, \(a_1 = -7\) and \(d = 3.1\).

Step 3: Find the n-th term formula

Substitute \(a_1 = -7\) and \(d = 3.1\) into the formula: \(a_n = -7 + (n-1) \, 3.1\). Simplify: \(a_n = -7 + 3.1n - 3.1 = 3.1n - 10.1\). Thus, the formula for the \(n\)th term is \(a_n = 3.1n - 10.1\).

Step 4: Calculate the 5th term

To find the 5th term, substitute \(n=5\) into the formula: \(a_5 = 3.1 \times 5 - 10.1\). Calculate: \(a_5 = 15.5 - 10.1 = 5.4\). Therefore, the 5th term is \(5.4\).

Step 5: Calculate the 10th term

To find the 10th term, substitute \(n=10\) into the formula: \(a_{10} = 3.1 \times 10 - 10.1\). Calculate: \(a_{10} = 31 - 10.1 = 20.9\). Therefore, the 10th term is \(20.9\).

Key Concepts

Common Difference

In an arithmetic sequence, each term after the first is formed by adding a constant value, known as the "common difference." This difference, often denoted by \(d\), is what sets arithmetic sequences apart from other types of sequences. For instance, in the sequence \(-7, -3.9, -0.8, 2.3, \ldots\), the common difference is the change between any two consecutive terms.

To calculate this, subtract the first term from the second:

• \((-3.9) - (-7) = 3.1\).

Thus, the common difference \(d\) is \(3.1\).
This value is added repeatedly to get from one term to the next in the sequence.

Understanding the common difference is essential because it directly influences how the sequence progresses and it's a key part of finding any term in the sequence using an arithmetic method.

n-th Term Formula

The n-th term formula is a powerful tool in arithmetic sequences. It helps you find any term in the sequence without listing all the previous terms. The general formula for the n-th term, \(a_n\), is:

• \(a_n = a_1 + (n-1) \cdot d\)

Here, \(a_1\) represents the first term of the sequence, and \(d\) is the common difference.

For our given sequence, \(a_1 = -7\) and \(d = 3.1\). Substituting these into our generic formula provides us a specific formula for this sequence:

• \(a_n = -7 + (n-1) \cdot 3.1\)
• \(a_n = 3.1n - 10.1\).

This formula allows you to plug any term number \(n\) into it to calculate the corresponding sequence term. Understanding and using this formula can save you time and effort, especially when dealing with large sequences.

Sequence Calculation

Sequence calculation is about using the tools and formulas we've discussed to find specific terms in an arithmetic sequence. Using the n-th term formula, \(a_n = 3.1n - 10.1\), you can find terms such as the 5th or the 10th term by substituting the term number \(n\) directly.

For example, to calculate the 5th term:

• Substitute \(n = 5\) into the formula: \(a_5 = 3.1 \times 5 - 10.1\).
• Do the arithmetic: \(a_5 = 15.5 - 10.1 = 5.4\).

Thus, the 5th term is \(5.4\).

Similarly, for the 10th term:

• Substitute \(n = 10\) into the formula: \(a_{10} = 3.1 \times 10 - 10.1\).
• Calculate: \(a_{10} = 31 - 10.1 = 20.9\).

Therefore, the 10th term is \(20.9\).
This method is straightforward and efficient for finding terms in an arithmetic sequence without needing to add the common difference repeatedly.