Problem 44 - Precalculus with Limits

Question

In Exercises 41 - 46, write the first five terms of the arithmetic sequence defined recursively. \( a_1 = 72, a_{n + 1} = a_n - 6 \)

Step-by-Step Solution

Answer

The first five terms of the arithmetic sequence are 72, 66, 60, 54, and 48.

Step 1: Identify first term

According to the given equation, the first term of the sequence is \( a_1 = 72 \) which is already provided in the problem.

Step 2: Apply recursive formula for the second term

The second term \( a_{n + 1} = a_n - 6 \) is obtained by subtracting 6 from the first term. Hence, \( a_2 = a_1 - 6 = 72 - 6 = 66 \).

Step 3: Apply recursive formula for third term

Subtract 6 from the second term to get the third term: \( a_3 = a_2 - 6 = 66 - 6 = 60 \).

Step 4: Apply recursive formula for fourth term

Similarly, the fourth term is \( a_4 = a_3 - 6 = 60 - 6 = 54 \).

Step 5: Apply recursive formula for fifth term

The fifth term is \( a_5 = a_4 - 6 = 54 - 6 = 48 \).

Step 6: Conclude the Result

The first five terms of this arithmetic sequence are 72, 66, 60, 54, and 48.

Key Concepts

Recursive Formula

A recursive formula is a way of defining each term in a sequence using the preceding term(s). In an arithmetic sequence, you determine the next term by adding or subtracting a constant value to the previous one. This constant is known as the common difference.
The beauty of a recursive formula lies in its simplicity. You only need to know one term (usually the first) and the rule to generate the others.

• For instance, in the exercise above, the recursive formula given is: \( a_{n + 1} = a_n - 6 \)
• Here, the sequence begins with \( a_1 = 72 \).
• This rule tells us to subtract 6 from the previous term to find the next one.

Using a recursive formula can save you time and effort, especially for long sequences.

Terms of a Sequence

The terms of a sequence are individual elements or members of the sequence. In an arithmetic sequence, each term is derived from the one before it by adding or subtracting a constant value.
Let's look closely at the terms in the given exercise:

• The first term, \( a_1 = 72 \), is given directly.
• The second term, \( a_2 = 66 \), results from the recursive rule: \( a_2 = 72 - 6 = 66 \).
• The third term, \( a_3 = 60 \), follows as: \( 66 - 6 = 60 \).
• The sequence continues: \( a_4 = 54 \) and \( a_5 = 48 \).

Understanding each term's position and value helps visualize the sequence's pattern.

Sequence Definition

A sequence can be understood as an ordered list of numbers, where each number is called a term. The sequence definition gives you the necessary rules or criteria to find all the terms.
The sequences can be arithmetic, geometric, or any other type.

• An arithmetic sequence, like our exercise example, has a common difference between consecutive terms.
• The original exercise defines the sequence using a recursive method, starting with the term \( a_1 = 72 \) and a rule to find the next terms: \( a_{n + 1} = a_n - 6 \).

Such sequences are prevalent in various mathematical problems, helping us model and solve progression-related queries efficiently.