Problem 55 - College Algebra with Corequisite Support

Question

For the following exercises, find the number of terms in the given finite arithmetic sequence. \(a=\left\\{\frac{1}{2}, 2, \frac{7}{2}, \ldots, 8\right\\}\)

Step-by-Step Solution

Answer

The sequence has 6 terms.

Step 1: Identify the First Term

The first term of the sequence, denoted as \(a_1\), is the first element of the given sequence. Hence, \(a_1 = \frac{1}{2}\).

Step 2: Determine the Common Difference

The common difference \(d\) in an arithmetic sequence is the difference between consecutive terms. Thus, \(d = 2 - \frac{1}{2} = \frac{3}{2}\).

Step 3: Identify the Last Term

The last term of the sequence, denoted as \(a_n\), is given as 8.

Step 4: Use the Formula for the nth Term

The nth term of an arithmetic sequence is given by the formula: \(a_n = a_1 + (n - 1) \cdot d\). Substitute the known values: \(8 = \frac{1}{2} + (n - 1) \cdot \frac{3}{2}\).

Step 5: Solve for n

Rearrange and solve for \(n\):1. Subtract \(\frac{1}{2}\) from both sides: \(8 - \frac{1}{2} = (n - 1) \cdot \frac{3}{2}\).2. Simplify: \(\frac{15}{2} = (n - 1) \cdot \frac{3}{2}\).3. Multiply both sides by \(\frac{2}{3}\): \(n - 1 = 5\).4. Add 1 to both sides: \(n = 6\).

Key Concepts

Finite Sequence

In mathematics, sequences can either be finite or infinite. A finite sequence has a definite number of terms. Each term is separated in an ordered manner, starting and ending with specific values. This means that in a finite sequence, you can easily identify the first and the last term. For instance, in the given sequence \(\left\{ \frac{1}{2}, 2, \frac{7}{2}, \ldots, 8 \right\}\), there are clear starting and stopping points.
In this particular sequence, the sequence begins at \(\frac{1}{2}\) and ends at 8. Knowing this allows us to carry out calculations with ease, as we have a clear grasp of where our sequence starts and ends.
So, whenever you're working with finite sequences, always start by identifying these boundaries to help you understand the scope of your work.

Common Difference

The common difference is a key concept in arithmetic sequences. It tells us how much we add (or subtract) to move from one term to the next. In our sequence, this is evident from examining the difference between the consecutive terms.

• For example, to find the common difference \(d\) here, observe the first two terms: \(2 - \frac{1}{2} = \frac{3}{2}\).

This consistent difference means we are dealing with an arithmetic sequence, where each term increases by \(\frac{3}{2}\) over the previous one.
Understanding the common difference helps to quickly identify any arithmetic sequence pattern. It’s the stepping stone that connects each term to the next, ensuring the sequence maintains its regularity and predictability.

Nth Term Formula

To explore any term in an arithmetic sequence, the nth term formula can be incredibly useful. This formula is expressed as \(a_n = a_1 + (n - 1) \cdot d\), allowing us to find any term position \(n\).
Let's break down what each part means:

• \(a_n\) is the nth term you’re looking for.
• \(a_1\) stands for the first term, which is \(\frac{1}{2}\) in our sequence.
• \(d\) is the common difference (\(\frac{3}{2}\)).
• \(n\) is the term position number in the sequence.

In practical applications, this equation allows us to plug in known values to find unknowns. For instance, using the last term value \(8\), we can rearrange and solve this formula to identify details about our sequence, like calculating the total number of terms.

Number of Terms

Finding the number of terms in a finite arithmetic sequence follows directly from using the nth term formula. After identifying the first term, the common difference, and the last term, here's how you proceed:

• Set the nth term formula with \(a_n\) as the last term (8 in this case).
• Substitute known values into \(8 = \frac{1}{2} + (n - 1) \cdot \frac{3}{2}\).
• Rearrange to solve for \(n\), representing the term position of the last term.
• Through algebra, you'll find \(n = 6\), indicating there are 6 terms in this sequence.

The calculation of \(n\) gives clarity on how the sequence is structured. Completing this step ensures you know not only where your sequence starts and ends, but also the full count of terms within it.