Problem 26 - Beginning and Intermediate Algebra

Question

For each arithmetic sequence, find \(a_{n}\) and then use \(a_{n}\) to find the indicated term. $$13,19,25,31,37, \ldots ; a_{30}$$

Step-by-Step Solution

Answer

The formula for the nth term of this arithmetic sequence is \(a_n = 13 + (n - 1)6\). Using this formula, we find that the 30th term, \(a_{30}\), equals 187.

Step 1: Identify the common difference

In an arithmetic sequence, the difference between consecutive terms is always the same. To find the common difference in this sequence, subtract the first term from the second term or any consecutive terms. In our case: \(d = 19 - 13 = 6\) The common difference is 6.

Step 2: Find the formula of the nth term

Now, we can find a formula for the nth term, \(a_n\). For arithmetic sequences, the formula for the nth term is: \[a_n = a_1 + (n - 1)d\] Where \(a_1\) is the first term, \(n\) is the term number, and \(d\) is the common difference. In our case, \(a_1 = 13\) and \(d = 6\). The formula for \(a_n\) is: \[a_n = 13 + (n - 1)6\]

Step 3: Find \(a_{30}\)

Now we use the formula to find \(a_{30}\), the 30th term in the arithmetic sequence. Simply replace \(n\) with 30 in the formula and solve. \[a_{30} = 13 + (30 - 1)6\] \[a_{30} = 13 + (29)6\] \[a_{30} = 13 + 174\] \[a_{30} = 187\] The 30th term, \(a_{30}\), is 187.

Key Concepts

Common Difference

In the fascinating world of arithmetic sequences, the **common difference** serves as the consistent gap between any two consecutive terms. Simply put, this value remains identical as you move from one term to the next within the sequence. To find it, subtract the first term from the second term. In our example sequence **13, 19, 25, 31, 37, ...**, you take 19 (the second term) and subtract 13 (the first term):

• Common difference, \(d\) = 19 - 13 = 6

Thus, every time you step forward in this sequence, you increase by 6.

nth Term Formula

The **nth term formula** is a cornerstone in understanding arithmetic sequences. It allows you to find any term in a sequence without listing all the preceding terms. This formula is expressed as:

• \(a_n = a_1 + (n - 1)d\)

Here:

• \(a_n\) is the term you want to find.
• \(a_1\) is the first term of the sequence.
• \(n\) is the position of the term in the sequence.
• \(d\) is the common difference.

Using our example with \(a_1 = 13\) and \(d = 6\), the formula becomes:

• \(a_n = 13 + (n - 1) \times 6\)

This shortcut greatly simplifies the process of finding any term in the sequence.

Arithmetic Sequence Example

Let's apply our knowledge of the common difference and nth term formula to an **arithmetic sequence example**. Consider the sequence: **13, 19, 25, 31, 37, ...**.In this case, we've already established:

• First term, \(a_1\) = 13
• Common difference, \(d\) = 6

To find a particular term such as the 30th term, we use our nth term formula. We know:

• \(n = 30\)

Substitute these values into the formula:

• \(a_{30} = 13 + (30 - 1) \times 6\)

So:

• \(a_{30} = 13 + 174 = 187\)

Therefore, the 30th term of the sequence is 187.

Sequence Term Calculation

**Sequence term calculation** allows us to pinpoint any specific term by plugging values into the generic nth term formula. Remember, the formula for an arithmetic sequence is:

• \(a_n = a_1 + (n - 1)d\)

Using this, let's calculate the stated term, the 30th, in our example sequence.To summarize our computation steps:

• Identify \(a_1 = 13\), \(d = 6\)
• Select \(n = 30\)
• Plug into the formula: \(a_{30} = 13 + (30 - 1) \times 6\)

Perform the math:

• \(a_{30} = 13 + 174 = 187\)

By following these simple steps, you can effortlessly find any term in an arithmetic sequence, illustrating the beauty and utility of these mathematical concepts.