Problem 41 - Beginning and Intermediate Algebra

Question

Find the number of terms in each arithmetic sequence. $$8,13,18,23, \dots, 63$$

Step-by-Step Solution

Answer

There are \(12\) terms in the given arithmetic sequence: \(8, 13, 18, 23, \dots, 63\).

Step 1: Identify the first term, common difference, and the last term

In the arithmetic sequence given, we can identify the following values: - The first term (a₁): 8 - Common difference (d): 13 - 8 = 5 - The last term (aₙ): 63

Step 2: Apply the arithmetic sequence formula

To find the number of terms (n) in the arithmetic sequence, we use the formula: \[aₙ = a₁ + (n - 1)d\] where aₙ is the last term, a₁ is the first term, d is the common difference, and n is the number of terms.

Step 3: Substitute the values and solve for n

Plug the values we obtained in step 1 into the formula: \[63 = 8 + (n - 1) \times 5\]

Step 4: Simplify and solve for n

To solve for n, follow these steps: 1. Distribute the 5 to the n-1: \[63 = 8 + 5n - 5\] 2. Combine like terms: \[63 = 3 + 5n\] 3. Subtract 3 from both sides: \[60 = 5n\] 4. Divide both sides by 5: \[n = 12\]

Step 5: Interpret the result

There are 12 terms in the given arithmetic sequence: \(8, 13, 18, 23, \dots, 63\).

Key Concepts

First Term

In an arithmetic sequence, the first term serves as the starting point. It is generally represented by the symbol \(a_1\). This term sets the pace for the entire sequence, and just like laying the first brick, it is crucial for the structure that follows. In our given problem, the first term (\(a_1\)) is 8. Thus, the sequence begins with the number 8. Each subsequent term is derived from adding a constant difference to the first term. Understanding the first term is essential because it allows us to calculate the other components of the arithmetic sequence, such as the common difference and the subsequent terms.

Common Difference

The common difference in an arithmetic sequence is the amount added to each term to produce the next term. It is symbolized by \(d\) and remains consistent across the terms of the sequence. This fixed difference is what makes an arithmetic sequence linear in its progression. For the given sequence \(8, 13, 18, 23, \dots, 63\), the common difference is calculated by subtracting the first term from the second term: \(13 - 8 = 5\). This means that each term is 5 units more than the previous one.

• First term: 8
• Second term: 13
• Third term: 18

Understanding the common difference allows you to generate further terms and is also integral in applying the arithmetic sequence formula for finding the number of terms.

Number of Terms

The number of terms in an arithmetic sequence refers to how many individual elements or numbers are there from the first term to the last. To solve for this, one utilizes the arithmetic sequence formula:

To find the number of terms \(n\), we use the formula \(a_n = a_1 + (n - 1)d\) where:

• \(a_n\) is the last term
• \(a_1\) is the first term
• \(d\) is the common difference

In our case, \(8, 13, 18, 23, \dots, 63\), we replace these with \(a_1 = 8\), \(d = 5\), and \(a_n = 63\) to find \(n\). By rearranging and solving the equation, \(n\) can be found, revealing there are 12 terms in the sequence.

Arithmetic Sequence Formula

The arithmetic sequence formula is a key mathematical tool used to find various properties of a sequence, such as the number of terms or a particular term. It is given by:

\[a_n = a_1 + (n - 1)d\]
This equation relates the last term \(a_n\) of the sequence to the first term \(a_1\), with \(d\) being the common difference and \(n\) the number of terms. This formula is particularly useful because it enables finding unknown variables when the rest are provided. In solving the exercise, this formula was crucial for determining the number of terms using the known first term, common difference, and last term. By substituting the known values into the formula and solving for \(n\), students can grasp how the sequence expands. It is a simple yet powerful rule that students will encounter in various mathematical applications.