Problem 54 - College Algebra with Corequisite Support

Question

For the following exercises, find the number of terms in the given finite arithmetic sequence. \(a=\\{1.2,1.4,1.6, \ldots, 3.8\\}\)

Step-by-Step Solution

Answer

There are 14 terms in the sequence.

Step 1: Identify the First Term (a₁)

The first term of the sequence is given as 1.2. This value is denoted as \( a_1 = 1.2 \).

Step 2: Determine the Common Difference (d)

To find the common difference, subtract the first term from the second term: \( d = 1.4 - 1.2 = 0.2 \).

Step 3: Identify the Last Term (a_n)

The last term of the sequence is given as 3.8, denoted as \( a_n = 3.8 \).

Step 4: Use the Formula to Find the Number of Terms (n)

The formula for the nth term of an arithmetic sequence is \( a_n = a_1 + (n - 1) \, d \). Substitute the values: \( 3.8 = 1.2 + (n - 1) \, 0.2 \).

Step 5: Solve for n

Rearrange the equation to solve for \( n \):1. Subtract 1.2 from both sides: \( 3.8 - 1.2 = 0.2(n - 1) \).2. Simplify: \( 2.6 = 0.2(n - 1) \).3. Divide both sides by 0.2: \( \frac{2.6}{0.2} = n - 1 \).4. Simplify: \( 13 = n - 1 \).5. Add 1 to both sides: \( n = 14 \).

Step 6: Conclusion

The number of terms in the sequence is 14.

Key Concepts

Number of Terms

In an arithmetic sequence, the number of terms refers to how many elements or terms are in the given sequence. To determine this, we use a specific formula. The general formula for the nth term of an arithmetic sequence is \( a_n = a_1 + (n-1)d \), where:

• \( a_n \) is the last term of the sequence.
• \( a_1 \) is the first term.
• \( d \) is the common difference between terms.
• \( n \) is the number of terms.

Substitute the known values into the formula to solve for \( n \). Rearrange the terms in the equation to isolate \( n \) on one side. In the provided exercise, this leads us to find that \( n = 14 \). Therefore, this sequence contains 14 terms in total.

Common Difference

The common difference in an arithmetic sequence is a critical component that dictates the pattern of the sequence. It is the constant amount that each term increases or decreases from the previous term. To find it, subtract the first term from the second term. For example, in our sequence:

• First term \( a_1 = 1.2 \)
• Second term = 1.4

The common difference \( d \) is calculated as \( 1.4 - 1.2 = 0.2 \). This value is the same for all pairs of successive terms in the sequence. Consistently applying this difference helps us predict subsequent terms in the sequence with ease.

First Term

The first term in an arithmetic sequence is the initial starting point or the first element of the list of numbers in the sequence. It is often denoted as \( a_1 \). Understanding the first term is essential because it significantly impacts the calculation of any other term in the sequence.
In our given sequence, the first term is \( a_1 = 1.2 \). This value sets the stage for the entire sequence. By knowing the first term, you can apply the common difference repeatedly to find the rest of the terms. It provides a baseline reference when using the arithmetic sequence formula.

Last Term

The last term of an arithmetic sequence marks the end of the sequence. It is denoted by \( a_n \) and plays a vital role in determining the total length of the sequence. The number of terms (\( n \)) in the sequence can be calculated using this term along with the first term and the common difference.
For the sequence in the exercise, the last term is given as \( a_n = 3.8 \). By knowing the last term, you can verify the completion of calculations and ensure the sequence does not accidentally extend past the desired range. It serves as a check in the final application of the arithmetic sequence formula.