Problem 26 - College Algebra with Corequisite Support

Question

For the following exercises, use the recursive formula to write the first five terms of the arithmetic sequence. $$ a_{1}=39 ; \quad a_{n}=a_{n-1}-3 $$

Step-by-Step Solution

Answer

The first five terms of the sequence are 39, 36, 33, 30, and 27.

Step 1: Identify the Given Values

We are given the first term of the arithmetic sequence, \(a_1 = 39\), and the recursive formula \(a_n = a_{n-1} - 3\). This recursive formula tells us how to find each subsequent term in the sequence by subtracting 3 from the previous term.

Step 2: Calculate the Second Term

Using the recursive formula, find the second term \(a_2\) by substituting \(a_1\):\[a_2 = a_1 - 3 = 39 - 3 = 36\]

Step 3: Calculate the Third Term

Use the recursive formula to find the third term \(a_3\) by substituting \(a_2\):\[a_3 = a_2 - 3 = 36 - 3 = 33\]

Step 4: Calculate the Fourth Term

Use the recursive formula to find the fourth term \(a_4\) by substituting \(a_3\):\[a_4 = a_3 - 3 = 33 - 3 = 30\]

Step 5: Calculate the Fifth Term

Use the recursive formula to find the fifth term \(a_5\) by substituting \(a_4\):\[a_5 = a_4 - 3 = 30 - 3 = 27\]

Step 6: List the First Five Terms

Now that we have calculated each term using the recursive formula, we list the first five terms: \(a_1 = 39\), \(a_2 = 36\), \(a_3 = 33\), \(a_4 = 30\), and \(a_5 = 27\).

Key Concepts

Understanding a Recursive Formula

A recursive formula allows us to generate terms of a sequence based on preceding ones. In the context of an arithmetic sequence, this formula is quite straightforward. We are given the first term, denoted as \(a_1\), and a rule for finding each subsequent term. In this particular example, we start with \(a_1 = 39\) and use the formula \(a_n = a_{n-1} - 3\). This instructs us to take the previous term and subtract 3 to find the next term.

Recursive formulas are immensely useful because they provide a simple way to generate terms without needing to recognize the entire pattern outright. As long as you know the initial term and the rule, you can continue the sequence indefinitely.

Calculating the First Five Terms

With a clear understanding of the recursive formula \(a_n = a_{n-1} - 3\) in mind, determining the first five terms of the sequence becomes systematic. We begin with the initial term, \(a_1 = 39\), and apply the formula repeatedly:

• First Term: \(a_1 = 39\)


• Second Term: \(a_2 = 39 - 3 = 36\)


• Third Term: \(a_3 = 36 - 3 = 33\)


• Fourth Term: \(a_4 = 33 - 3 = 30\)


• Fifth Term: \(a_5 = 30 - 3 = 27\)

By consistently applying the recursive formula, we can quickly list the first five terms of this arithmetic sequence as 39, 36, 33, 30, and 27.

Exploring Subsequent Term Calculation

The subsequent term calculation in an arithmetic sequence involves using the recursive formula. Each new term can be computed by modifying the prior term as defined by the formula. In this sequence, each term is obtained by decreasing the previous term by 3. This process might seem repetitive but is valuable for automation in sequences.

Whenever you encounter a recursive formula, remember:

• Identify the starting term, in our case, \(a_1 = 39\).
• Apply the rule or pattern, here it’s \(a_n = a_{n-1} - 3\).
• Repeat the process for each subsequent term as needed.

Through this method, you can construct any number of terms in an arithmetic sequence efficiently and without error.