Problem 31 - Contemporary Precalculus

Question

Find the first five terms of the recursively defined sequence. $$\begin{array}{ll}a_{1}=1, a_{2}=-2, a_{3}=3, & \text { and } \\\a_{n}=a_{n-1}+a_{n-2}+a_{n-3} & \text { for } n \geq 4\end{array}$$

Step-by-Step Solution

Answer

Answer: The first five terms of the sequence are 1, -2, 3, 2, and 3.

Step 1: Understand the provided terms

The first three terms of the sequence are given as: $$a_1 = 1, \; a_2 = -2,\; a_3 = 3$$

Step 2: Apply the general rule equation to find the 4th term (a4)

We have the formula: $$a_n = a_{n-1} + a_{n-2} + a_{n-3} \text{ for } n\geq 4$$ To find a4, let n = 4: $$a_4 = a_{4-1} + a_{4-2} + a_{4-3} = a_3 + a_2 + a_1$$ By substituting the values, we get: $$a_4 = 3 + (-2) +1 = 2 $$

Step 3: Apply the general rule equation to find the 5th term (a5)

To find a5, let n = 5: $$a_5 = a_{5-1} + a_{5-2} + a_{5-3} = a_4 + a_3 + a_2$$ By substituting the values, we get: $$a_5 = 2 + 3 +(-2) = 3$$

Step 4: Listing the first five terms of the sequence

We have found all the first five terms of the sequence: $$a_1 = 1, \; a_2 = -2, \; a_3 = 3, \; a_4 = 2, \; a_5 = 3$$

Key Concepts

Sequence Definition

In mathematics, a sequence is a list of numbers in a specific order. These numbers are called terms. Sequences can be defined using a particular rule or formula. A common way to define a sequence is through a recursive formula.

• A recursive formula gives us instructions to find any term based on one or more previous terms.
• In the given exercise, the sequence is defined recursively with the initial terms provided and a rule to find the subsequent terms.

Understanding the sequence definition is crucial. It serves as the starting point for calculating each term. The initial terms provided (like in the exercise) set up how the sequence begins. Then, the recursive formula guides the calculation of future terms.

Term Calculation

Once we know how a sequence is defined, calculating each term becomes the next step.

• For a recursive sequence, we repeatedly apply the formula to generate terms.
• In our example, the sequence starts with the initial terms \(a_1 = 1\), \(a_2 = -2\), and \(a_3 = 3\).
• The formula \(a_n = a_{n-1} + a_{n-2} + a_{n-3}\) is used to find further terms when \(n \geq 4\).

To find subsequent terms, plug previously calculated values back into the formula. For example, the fourth term is calculated by \(a_4 = a_3 + a_2 + a_1\), substituting known values, \(a_4 = 3 + (-2) + 1 = 2\). Similarly, for \(a_5\), substitute the latest values, resulting in \(a_5 = 2 + 3 + (-2) = 3\). Experimenting with these calculations is essential to understanding how new values are generated from earlier terms.

Patterns in Sequences

Recognizing patterns in sequences helps with predicting future terms and understanding the sequence's behavior.

• Observing changes between terms reveals regularity or irregularity in a sequence.
• In our exercise, notice how each subsequent term relies heavily on its predecessors.
• The recursive formula shows how past values influence each new term, creating a cascading pattern.

By analyzing the first five terms calculated \((1, -2, 3, 2, 3)\), one might deduce that the sequence doesn't settle into a simple arithmetic or geometric pattern. Instead, it dynamically changes with each inclusion of previous terms. Detecting these patterns doesn't just aid in calculations but also deepens our understanding of how sequences evolve.