Problem 13 - Algebra 2

Question

Write a recursive formula for each sequence. Then find the next term. $$ 43,41,39,37,35, \ldots $$

Step-by-Step Solution

Answer

The recursive formula for the sequence is \(a_{n} = a_{n-1} - 2\), for \(n > 1\), \(a_{1} = 43\). The next term in the sequence is \(33\).

Step 1: Identify the Recursive Pattern

Observe the given sequence \(43, 41, 39, 37, 35,...\). We can immediately see that the next term is always 2 less than the current term. The common difference here is \(-2\). An arithmetic sequence can always described in a recursive way using the formula \(a_{n} = a_{n-1} + d\), where \(a_{n}\) represents the nth term, \(a_{n-1}\) represents the previous term, and \(d\) represents the common difference.

Step 2: Write the Recursive Formula

Now that we've identified the pattern, we can write out the recursive formula. In the case of our sequence, where each term is calculated by subtracting 2 from the previous term, our recursive formula becomes \(a_{n} = a_{n-1} - 2\), for \(n > 1\) where \(a_{1} = 43\).

Step 3: Find the Next Term Using the Recursive Formula

In order to determine the next term of the sequence, substitute the last given term (35) into the recursive formula. So, \(a_{6} = a_{5} - 2 = 35 - 2 = 33\). Thus, the next term in the sequence is 33.

Key Concepts

Recursive Formula

A recursive formula for an arithmetic sequence is a powerful tool that helps us determine the terms of a sequence step-by-step, based on the relation between consecutive terms. In the context of sequences, a recursive formula builds the entire sequence using previous terms, acting like a chain that connects each term to the one before it. For an arithmetic sequence, this formula typically looks like this:

• \( a_{n} = a_{n-1} + d \)
• A key element of this formula is \(a_{n}\), representing the nth term.
• Next comes \(a_{n-1}\), which is the previous term in the sequence.
• And finally, \(d\), the common difference, which describes how we move from one term to the next.

By following this simple rule, you can generate the entire sequence once you know the first term. In our specific example of the sequence \(43, 41, 39, 37, 35, \, \ldots\), the recursive formula quickly reveals itself when we observe how each term is formed by subtracting 2 from the preceding term. Here, the initial term, \(a_1\), is 43, and the recursive formula becomes \(a_{n} = a_{n-1} - 2 \). This formula helps us continue the sequence indefinitely, maintaining its structure and pattern.

Common Difference

Understanding the common difference is crucial for identifying arithmetic sequences, as it dictates how the sequence progresses. The common difference (\(d\)) is a constant interval between consecutive terms in an arithmetic sequence, which stays consistent throughout. To find this value, look at any pair of consecutive terms, and calculate their difference by subtracting the earlier term from the later one. This simple subtraction reveals the common difference. In our example sequence \(43, 41, 39, 37, 35, \, \ldots\), the difference between each term is

• \(41 - 43 = -2\),
• \(39 - 41 = -2\),
• \(37 - 39 = -2\),
• \(35 - 37 = -2\).

All of these calculations confirm that the common difference is consistently \(-2\). This negative value indicates that the sequence is decreasing, with each term being 2 less than the one before it. Knowing the common difference allows us to use the recursive formula effectively, and anticipate future terms in the sequence.

Sequence Patterns

Recognizing sequence patterns is essential for understanding arithmetic sequences and predicting future terms. Sequence patterns refer to the ongoing rules and relationships that connect consecutive terms within a sequence. These patterns involve regular differences, which define how the sequence progresses. In arithmetic sequences, the pattern is characterized by its constant common difference, which means that the change between any two consecutive terms remains the same. By identifying this pattern, it becomes significantly easier to establish a recursive formula and predict unknown terms.In the sequence \(43, 41, 39, 37, 35, \, \ldots\), the pattern is straightforward: each term is derived by subtracting 2 from the previous one. Once you've grasped this pattern, the route to the next few terms becomes clear, for example

• If the last term given is 35, then subtract 2, resulting in the next term, 33.
• If you continue applying this pattern, the subsequent terms are 31, then 29, and so forth.

Recognizing and understanding these patterns not only help solve problems in a structured way but also enhance mathematical intuition when dealing with various sequences.