Problem 19 - Algebra 2 and Trigonometry

Question

Write a recursive definition for an arithmetic sequence with a common difference of \(-3\)

Step-by-Step Solution

Answer

The recursive definition is: Base case \(a_1 = c\); Recursive formula \(a_{n+1} = a_n - 3\).

Step 1: Understanding the Components

An arithmetic sequence is a sequence of numbers such that the difference between consecutive terms is constant. This difference is called the common difference. In our case, the common difference is \(-3\).

Step 2: Defining the Base Case

To write a recursive definition, we need a base case, which is the first term of our sequence. Let's denote the first term as \(a_1 = c\), where \(c\) is some constant. This is where our sequence starts.

Step 3: Writing the Recursive Formula

In a recursive definition, each term is expressed in terms of the previous term. So for our arithmetic sequence, each term is \(a_{n+1} = a_n - 3\). This shows that each subsequent term is the previous term minus the common difference of \(-3\).

Step 4: Complete Recursive Definition

Putting it all together, we have:- Base case: \(a_1 = c\)- Recursive formula: \(a_{n+1} = a_n - 3\) for \(n \geq 1\)

Step 5: Example

If we let \(c = 5\), then the sequence starts with \(5\), and the next terms are:- \(a_2 = 5 - 3 = 2\)- \(a_3 = 2 - 3 = -1\)- \(a_4 = -1 - 3 = -4\)Thus, the sequence is \(5, 2, -1, -4, \ldots\)

Key Concepts

Recursive Definition

In mathematics, a recursive definition is like a set of instructions that tells you how to create something step-by-step. With regard to sequences like arithmetic sequences, it explains how to generate each term in the sequence using the previous term. To establish a recursive definition for an arithmetic sequence, you need two main components: the base case and the recursive formula.

• The base case specifies the starting point of the sequence, which is the first term.
• The recursive formula explains how you can find the next term using the current term.

In the context of arithmetic sequences, the recursive formula uses the common difference, a constant value that you add (or subtract) to obtain the next term. This makes recursive definitions particularly suited for sequences with a predictable and repeating pattern, such as arithmetic sequences with a constant incremental change.

Common Difference

The common difference is a vital component in understanding arithmetic sequences. It is the consistent difference between consecutive terms in the sequence. This means that if you subtract any term from the term that follows it, you will always get the same number, known as the common difference. In our example, the common difference is \(-3\).
The significance of this value is that it defines the uniform change in the sequence. If the common difference is positive, the sequence increases as you proceed from one term to the next. Conversely, if the common difference is negative, the sequence decreases.

• For instance, in the sequence given in our problem, the common difference is \(-3\), meaning each term is \(-3\) less than the previous one.
• It's what gives the sequence its characteristic linearity and predictability.

Understanding the common difference allows you to predict subsequent terms without writing out the whole sequence.

Base Case

The concept of a base case is foundational in recursive definitions. It tells us where the sequence starts. Think of the base case as the seed you plant to grow the sequence. For a recursive formula to work correctly, it must have a specified starting point, which is what the base case provides.
In our example, the base case is given as \( a_1 = c \), where \ c \ is a constant you choose according to the context of the problem. This means that the first term of the sequence is \ c \. For every arithmetic sequence, there is always one unique base case that establishes the sequence's initial condition.

• Without a base case, the recursive definition is incomplete because there is no term to start with.
• The base case allows us to start applying the recursive formula to obtain the rest of the sequence.

Once the base case is defined, you can use the recursive formula to generate further terms by repeatedly applying the common difference.