Problem 28 - Algebra for College Students

Question

Write a formula for the general term of each infinite sequence. \(5,7,9,11,13, \ldots\)

Step-by-Step Solution

Answer

General term: \(a_n = 2n + 3\)

Step 1 - Identify the Pattern

Determine the pattern in the sequence: the difference between consecutive terms is the same. For the given sequence (5, 7, 9, 11, 13,...), the difference between each term is 2.

Step 2 - Determine the First Term

Identify the first term of the sequence. Here, the first term (a) is 5.

Step 3 - Write the General Term Formula

The general term of an arithmetic sequence can be written as: \[ a_n = a + (n-1)d \] where \(a\) is the first term and \(d\) is the common difference. Plugging in our values from steps 1 and 2: \[ a_n = 5 + (n-1) \times 2 \]

Step 4 - Simplify the Formula

Simplify the expression obtained in Step 3: \[ a_n = 5 + 2n - 2 \] \[ a_n = 2n + 3 \]

Key Concepts

General Term Formula

In arithmetic sequences, every term after the first one is created by adding a constant value to the previous term. This constant value is known as the common difference. The general term formula of an arithmetic sequence helps us quickly find any term in the sequence without listing all the previous ones. The formula looks like this:
\[ a_n = a + (n-1)d \]
Here, \(a_n\) is the nth term of the sequence, \(a\) is the first term, \(d\) is the common difference, and \(n\) is the position of the term in the sequence.
Using this formula, you can find any term in the sequence by plugging in the values for \(a\), \(d\), and \(n\). This saves time and simplifies the process.

Common Difference

The common difference is a key concept in understanding arithmetic sequences. It's the amount you add each time to get to the next term. Finding this difference is straightforward:

• Subtract the first term from the second term.
• For example, in the sequence 5, 7, 9, 11, 13, the common difference is 2 (7 - 5).

Once the common difference \(d\) is known, you can use it in the general term formula to find any term in the sequence.
To double-check your common difference:

• Subtract other consecutive terms (e.g., 9 - 7, 11 - 9) to ensure it’s consistent.

Consistency confirms that you are indeed dealing with an arithmetic sequence. Remember, each term is simply the previous term plus the common difference.

First Term

The first term of an arithmetic sequence (\(a\)) is where everything starts. It’s crucial for forming the general term formula. In our given sequence (5, 7, 9, 11, 13), the first term is 5. Knowing this term allows you to accurately establish the rest of the sequence by repeatedly adding the common difference.
Here are some helpful insights:

• The first term is often denoted by \(a\).
• It sets the starting point, and the pattern carried forth by adding the common difference creates the entire sequence.
• When plugging into the general term formula \( a_n = a + (n-1)d \), \(a\) is the value you begin with.

So, in practice, if you’re asked to find the 10th term in the sequence (5, 7, 9, 11, 13), you would start with 5 (the first term), and use the common difference of 2 in your calculations. This systematic approach makes dealing with sequences both efficient and manageable.