Problem 13 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 13-18: Find the specified term of the arithmetic sequence that has the two given terms. $$ a_{12} ; \quad a_{1}=9.1, \quad a_{2}=7.5 $$

Step-by-Step Solution

Answer

The 12th term, \(a_{12}\), of the sequence is \(-8.5\).

Step 1: Understand the Problem

We are given an arithmetic sequence and need to find the 12th term, \(a_{12}\). We know \(a_1 = 9.1\) and \(a_2 = 7.5\). In an arithmetic sequence, each term is obtained by adding a common difference \(d\) to the previous term.

Step 2: Determine the Common Difference

The common difference \(d\) can be found using the formula: \(d = a_2 - a_1\). We have \(a_2 = 7.5\) and \(a_1 = 9.1\). Substitute these values to get:\[d = 7.5 - 9.1 = -1.6\]

Step 3: Use the Term Formula for Arithmetic Sequences

The formula for the \(n\)-th term in an arithmetic sequence is given by:\[a_n = a_1 + (n - 1) \cdot d.\]We need to find \(a_{12}\), so substitute \(n = 12\), \(a_1 = 9.1\), and \(d = -1.6\) into the formula.

Step 4: Substitute into the Term Formula and Solve

Plug the values into the formula:\[a_{12} = 9.1 + (12 - 1) \cdot (-1.6)\]Simplify the expression:\[a_{12} = 9.1 + 11 \cdot (-1.6)\]Calculate \(11 \cdot (-1.6) = -17.6\) and find:\[a_{12} = 9.1 - 17.6 = -8.5.\]

Step 5: Verify the Result

Check the calculation once more: substitute \(n = 12\) back into the formula:\[a_{12} = 9.1 + 11 \times (-1.6) = 9.1 - 17.6 = -8.5.\]The calculation confirms that the answer is correct.

Key Concepts

Common Difference

In arithmetic sequences, the **common difference** is the consistent interval between consecutive terms. It's what makes an arithmetic sequence unique: every term increases or decreases by the same amount. To find it, you need the first two terms of your sequence, which in this case are given as \(a_1 = 9.1\) and \(a_2 = 7.5\).

The common difference \(d\) is calculated as follows:

• Subtract the first term from the second term.
• This gives us: \[d = a_2 - a_1 = 7.5 - 9.1 = -1.6.\]

A negative common difference, as we see here, indicates that the sequence is decreasing.

n-th Term Formula

The **n-th term formula** of an arithmetic sequence allows you to find any term in the sequence without listing all previous terms. This formula is essential when trying to find terms far into the sequence. The formula is:

• \[a_n = a_1 + (n - 1) \cdot d.\]

Here's what each part represents:

• \(a_n\) is the term you want to find.
• \(a_1\) is the first term in the sequence.
• \(n\) is the term's position in the sequence.
• \(d\) is the common difference.

Using this formula, you can plug in any position \(n\) to find the corresponding term. It's a straightforward substitution that offers great computational convenience.

Sequence Calculation

Performing a **sequence calculation** using the n-th term formula can help find specific terms quickly and efficiently. Let’s calculate the 12th term of the sequence using the information from our previous sections.

First, plug the known values into the n-th term formula:

• Given: \(a_1 = 9.1\), \(d = -1.6\), and \(n = 12\).
• The formula to use: \[a_{12} = 9.1 + (12 - 1) \cdot (-1.6).\]
• Calculate: \[a_{12} = 9.1 + 11 \times (-1.6) = 9.1 - 17.6.\]
• The result: \(a_{12} = -8.5.\)

This example demonstrates how to substitute values into the formula effectively. It highlights the predictability that the structure of arithmetic sequences provides.