Problem 38 - College Algebra with Corequisite Support

Question

For the following exercises, write a recursive formula for the given arithmetic sequence, and then find the specified term. \(a=\\{7,4,1, \ldots\\} ;\) Find the \(17^{\text {th }}\)

Step-by-Step Solution

Answer

The 17th term is -41.

Step 1: Identify the First Term and Common Difference

In the sequence provided, the first term, denoted as \(a_1\), is 7. The sequence is arithmetic, so each term is derived by adding a constant value to the previous term. We calculate the common difference \(d\) by subtracting the first term from the second term: \(d = 4 - 7 = -3\).

Step 2: Write the Recursive Formula

A recursive formula expresses each term in the sequence based on the preceding term. For an arithmetic sequence, the recursive formula is generally given by \(a_n = a_{n-1} + d\), where \(d\) is the common difference. For this sequence, the recursive formula is \(a_n = a_{n-1} - 3\) with the initial condition \(a_1 = 7\).

Step 3: Determine the 17th Term Using the Recursive Formula

To find the 17th term, \(a_{17}\), we repeatedly apply the recursive formula starting from the first term. However, it is easier to use the explicit formula to skip straight to any specific term. The explicit formula for an arithmetic sequence is \(a_n = a_1 + (n-1)d\). Substituting the known values (\(a_1 = 7\), \(d = -3\), and \(n = 17\)): \[a_{17} = 7 + (17-1)(-3)\]Simplify:\[a_{17} = 7 + 16(-3)\]\[a_{17} = 7 - 48\]\[a_{17} = -41\].

Key Concepts

Recursive Formula

A recursive formula helps you find each term of an arithmetic sequence using the one before it. When you know the common difference and the first term, you can create a recursive formula. This is useful when you're moving from one term to the next step-by-step.

The general form of a recursive formula for an arithmetic sequence is:

• \( a_n = a_{n-1} + d \)

where:

• \( a_n \) is the term you want to find.
• \( a_{n-1} \) is the previous term.
• \( d \) is the common difference.

For instance, if your sequence starts at 7 and the difference is -3, the recursive rule becomes \( a_n = a_{n-1} - 3 \). This rule allows you to write down terms consecutively as you work through the series, which can be a bit tedious if you need a term far away from the start.

Common Difference

Understanding the common difference is crucial in arithmetic sequences. It is the constant gap between every pair of consecutive terms in the sequence.

To find the common difference, subtract any term in the sequence from the term that follows it. For the given sequence \( \{7, 4, 1, \ldots\} \), the common difference \( d \) is calculated as:

• \( d = 4 - 7 = -3 \)

This negative value indicates the sequence decreases with each step. The common difference guides how quickly the sequence grows or shrinks as you move from one term to the next. It's important for developing both recursive and explicit formulas.

Explicit Formula

The explicit formula offers a quick way to find any term in the sequence without working through each precedent term. This formula is particularly helpful when you need to find a term that is far along in the series.

The explicit formula for an arithmetic sequence is:

• \( a_n = a_1 + (n-1)d \)

where:

• \( a_n \) is the nth term you wish to find.
• \( a_1 \) is the first term in the sequence.
• \( d \) is the common difference.
• \( n \) is the term number you are targeting.

For example, if you wish to determine the 17th term for the sequence \( \{7, 4, 1, \ldots\} \) with a common difference of -3, then the calculation is:

• \( a_{17} = 7 + (17-1)(-3) = 7 - 48 = -41 \)

This formula saves time and effort, particularly for terms far away in the sequence, by providing a direct calculation method.