Problem 27 - Beginning and Intermediate Algebra

Question

For each arithmetic sequence, find \(a_{n}\) and then use \(a_{n}\) to find the indicated term. $$1, \frac{3}{2}, 2, \frac{5}{2}, 3, \dots ; a_{18}$$

Step-by-Step Solution

Answer

The 18th term of the arithmetic sequence can be found using the formula \(a_{n} = a_{1} + (n-1)d\), where \(a_{1} = 1\), \(d = \frac{1}{2}\), and \(n = 18\). Calculating \(a_{18}\), we get \(a_{18} = 1 + (17)\frac{1}{2} = \frac{19}{2}\). So, the 18th term is \(\frac{19}{2}\).

Step 1: Finding the common difference

To find the common difference, we can subtract the first term from the second term, or the second term from the third term, and so on: $$d = \frac{3}{2} - 1 = \frac{1}{2}$$ The common difference d is \(\frac{1}{2}\).

Step 2: Use the formula to find \(a_{n}\)

Now that we know the common difference, we can use the formula for the nth term of an arithmetic sequence: $$a_{n} = a_{1} + (n-1)d$$ We are given that \(a_{1} = 1\), \(d = \frac{1}{2}\), and we want to find the \(18th\) term (\(a_{18}\)), so we have: $$a_{18} = 1 + (18-1)\frac{1}{2}$$

Step 3: Calculate the value of \(a_{18}\)

Now, we'll solve for \(a_{18}\): $$a_{18} = 1 + (17)\frac{1}{2} = 1 + \frac{17}{2} = \frac{2+17}{2} = \frac{19}{2}$$ The 18th term of the given arithmetic sequence is \(\frac{19}{2}\).

Key Concepts

Common Difference

In arithmetic sequences, the common difference is a key element that connects one term to the next. It remains constant throughout the sequence, which means you add or subtract the same number to get from one term to the next. To find the common difference, you simply subtract any two consecutive terms.

Consider the arithmetic sequence:

• The first term is 1.
• The second term is \(\frac{3}{2}\).

To find the common difference \(d\), subtract the first term from the second term:
\[d = \frac{3}{2} - 1 = \frac{1}{2}\]
This tells us that each term in the sequence increases by \(\frac{1}{2}\). Knowing the common difference helps predict patterns and find specific terms within the sequence.

Nth Term Formula

The nth term formula is an essential tool when working with arithmetic sequences. It allows you to calculate any term of the sequence without listing all the preceding terms. The formula is given by:

• \(a_{n} = a_{1} + (n-1)d\)

In this formula:

• \(a_{n}\) is the nth term you're trying to find.
• \(a_{1}\) is the first term of the sequence.
• \(d\) is the common difference.
• \(n\) is the position of the term in the sequence.

For example, in the sequence \(1, \frac{3}{2}, 2, \frac{5}{2}, 3, \dots\), if we want to find the 18th term, \(a_{18}\), we would use:
\[a_{18} = 1 + (18-1)\left(\frac{1}{2}\right)\]
The nth term formula simplifies and streamlines finding terms in long sequences.

Finding Terms in a Sequence

Once you have the common difference and the nth term formula for an arithmetic sequence, finding specific terms becomes a straightforward task.

Let's apply what we know to find the 18th term of the sequence \(1, \frac{3}{2}, 2, \frac{5}{2}, 3, \dots\):

• The first term \(a_{1}\) is 1.
• The common difference \(d\) is \(\frac{1}{2}\).

We use the nth term formula:
\[a_{18} = 1 + (18-1)\left(\frac{1}{2}\right)\]
Solving that, we calculate:

• Subtract 1 from 18 to get 17.
• Multiply 17 by the common difference \(\frac{1}{2}\) to get \(\frac{17}{2}\).
• Add this to the first term: 1 + \(\frac{17}{2}\) = \(\frac{19}{2}\).

The 18th term is \(\frac{19}{2}\). By understanding how to find terms in a sequence, you can efficiently solve more complex sequence problems.