Question

In Exercises \(8-11\), find the \(5^{\text {th }}, 10^{\text {th }}, n^{\text {th }}\) term of the arithmetic sequences. $$ 5,7.2, \ldots $$

Short answer

The 5th term is 13.8, the 10th term is 24.8, and the general formula for the nth term is \(a_n = 5 + (n - 1)(2.2)\).

Step-by-step solution

Identify the common difference

The common difference (d) is the difference between consecutive terms in an arithmetic sequence. To find the common difference, subtract the first term from the second term: \(7.2 - 5 = 2.2\). So, the common difference is 2.2.

Use the formula for the nth term of an arithmetic sequence

The formula for finding the nth term of an arithmetic sequence is: \(a_n = a_1 + (n - 1)d\), where \(a_n\) is the nth term, \(a_1\) is the first term, n is the term number, and d is the common difference. In our case, the first term \(a_1\) is 5 and the common difference d is 2.2.

Find the 5th term

To find the 5th term, plug in the values into the formula: \(a_5 = 5 + (5 - 1)(2.2)\). Calculate the result: \(a_5 = 5 + (4)(2.2) = 5 + 8.8 = 13.8\). So, the 5th term is 13.8.

Find the 10th term

To find the 10th term, plug in the values into the formula: \(a_{10} = 5 + (10 - 1)(2.2)\). Calculate the result: \(a_{10} = 5 + (9)(2.2) = 5 + 19.8 = 24.8\). So, the 10th term is 24.8.

Find the nth term

To find the nth term, plug in the given values into the formula: \(a_n = 5 + (n - 1)(2.2)\). The general formula for the nth term of this arithmetic sequence is: \(a_n = 5 + (n - 1)(2.2)\). In summary, the 5th term is 13.8, the 10th term is 24.8, and the general formula for the nth term is \(a_n = 5 + (n - 1)(2.2)\).

Key concepts

nth term formula

To understand arithmetic sequences, it's crucial to grasp the concept of the "\(n^{th}\) term formula." This formula helps you find any term in the sequence without manually calculating each step. The formula is represented as: \[a_n = a_1 + (n - 1)d\]Where:

• \(a_n\) is the \(n^{th}\) term we want to find.
• \(a_1\) is the first term of the sequence.
• \(n\) is the term number.
• \(d\) is the common difference.

This formula simplifies the process of finding any term, as you just need to know the first term and the common difference. In our example, the first term \(a_1\) is 5, and the common difference \(d\) is 2.2. By plugging these values into the formula, you can calculate any term in the sequence. This method is not only efficient but ensures accuracy in finding terms far along in the sequence.

common difference

The common difference in an arithmetic sequence is what sets these sequences apart. It's the constant value that you add to each term to get to the next one. To find it, subtract the first term from the second term. For instance, with our sequence \(5, 7.2, \ldots\), the common difference \(d\) is calculated as: \[d = 7.2 - 5 = 2.2\]

• The common difference \(d\) is always the same between any two consecutive terms.
• This consistency is what classifies the sequence as "arithmetic."

Understanding the common difference helps you see the pattern in the sequence. Knowing \(d\) allows you to predict future sequence terms simply by adding \(d\) successively to any current known term.

sequence terms

Sequence terms are the individual elements that make up the sequence. In an arithmetic sequence, each term is determined based on its position and the common difference. For example, to find specific terms, you follow this method:

• 5th term: To find the 5th term, apply the \(n^{th}\) term formula, using \(n=5\). So, \(a_5 = 5 + (5 - 1)(2.2) = 13.8\).
• 10th term: Similarly, for the 10th term, you use \(n=10\). Thus, \(a_{10} = 5 + (9)(2.2) = 24.8\).
• General term (\(n^{th}\) term): For any other term, \(a_n = 5 + (n - 1)(2.2)\).

Understanding individual sequence terms and how to calculate them empowers you to explore and analyze sequences thoroughly. It helps establish patterns and predict future terms based on mathematical logic, rather than guesswork. This aspect is foundational for solving complex mathematical problems involving sequences.