Problem 30 - Beginning and Intermediate Algebra

Question

For each arithmetic sequence, find \(a_{n}\) and then use \(a_{n}\) to find the indicated term. $$a_{1}=-5, d=-7 ; a_{14}$$

Step-by-Step Solution

Answer

The 14th term of the arithmetic sequence is \(a_{14}=-96\).

Step 1: Write down the given information

We are given the following information: $$a_1 = -5, d = -7$$

Step 2: Use the formula for the nth term of an arithmetic sequence

The formula for the nth term of an arithmetic sequence is given by: $$a_n = a_1 + (n-1) d$$

Step 3: Plug the given values into the formula to find \(a_n\)

We can plug \(a_1\) and \(d\) into the formula to find \(a_n\) as follows: $$a_n = -5 + (n-1)(-7)$$

Step 4: Solve for the 14th term (\(a_{14}\))

Now that we have the formula for \(a_n\), we can plug in \(n = 14\) to find the 14th term (\(a_{14}\)) of the sequence: $$a_{14} = -5 + (14-1)(-7)$$

Step 5: Simplify and calculate

Simplify the expression and calculate \(a_{14}\): $$a_{14} = -5 + (13)(-7)$$ $$a_{14} = -5 - 91$$ $$a_{14} = -96$$ Hence, the 14th term of the arithmetic sequence is \(-96\).

Key Concepts

Understanding the nth Term Formula

To solve problems involving arithmetic sequences, it's important to understand the nth term formula. This formula helps us find any term in a sequence without having to manually count each step from the beginning. The formula for the nth term, represented as \( a_n \), is given by:\[ a_n = a_1 + (n-1)d \]Where:

• \( a_1 \) is the first term of the sequence.
• \( d \) is the common difference between consecutive terms.
• \( n \) is the position of the term in the sequence.

Using this formula effectively, we can calculate any term by just knowing the first term and the difference. This saves time and effort, especially with large sequences. For example, if we have \( a_1 = -5 \) and \( d = -7 \), we can easily find any term like \( a_{14} \) by substituting these values into the formula.

Arithmetic Sequence Examples in Context

Learning from examples is one of the best ways to grasp how arithmetic sequences work. Let's consider our example where \( a_1 = -5 \) and \( d = -7 \).This sequence begins at -5 and decreases by 7 with each subsequent term, resulting in the sequence:

• -5
• -12
• -19
• -26
• ...

You will notice that each term is \( 7 \) less than the one before. This consistent pattern shows the fixed nature of arithmetic sequences and how the common difference \( d \) is applied. By understanding this, anyone can quickly generate more terms from the sequence or even work backwards to find missing values.

Solving Arithmetic Sequence Problems

Solving arithmetic sequence problems usually revolves around applying the nth term formula to find specific terms. Let’s work through a problem step by step.

Given: \( a_1 = -5 \), \( d = -7 \), find \( a_{14} \).

Step 1: Use the nth term formula: \( a_n = a_1 + (n-1)d \).
Step 2: Insert the values into the formula: \( a_n = -5 + (n-1)(-7) \).
Step 3: Substitute \( n = 14 \): \( a_{14} = -5 + (14-1)(-7) \).
Step 4: Simplify: \( a_{14} = -5 + 13(-7) \).
Step 5: Calculate: \( a_{14} = -5 - 91 = -96 \).To solve these types of problems, make sure to correctly substitute and simplify the expressions. Practicing with different sequences will enhance your understanding and ability to solve any related problem quickly and confidently.