Problem 29 - Contemporary Precalculus

Question

Find the first five terms of the recursively defined sequence. $$a_{1}=2 \text { and } a_{n}=n-a_{n-1} \quad \text { for } n \geq 2$$

Step-by-Step Solution

Answer

Answer: The first five terms of the sequence are: $$2$$, $$0$$, $$3$$, $$1$$, and $$4$$.

Step 1: Find the value of a_2

Given, $$a_{1}=2$$. To find the value of $$a_2$$, plug in $$n=2$$ into the recursive formula: $$a_{2}=2-a_{1}=2-2=0$$

Step 2: Find the value of a_3

Using the value of $$a_2$$ we found in Step 1 and the recursive formula: $$a_{3}=3-a_{2}=3-0=3$$

Step 3: Find the value of a_4

Using the value of $$a_3$$ we found in Step 2 and the recursive formula: $$a_{4}=4-a_{3}=4-3=1$$

Step 4: Find the value of a_5

Using the value of $$a_4$$ we found in Step 3 and the recursive formula: $$a_{5}=5-a_{4}=5-1=4$$

Step 5: Write down all the first five terms

Now that we have all the values, we can write down the first five terms of the sequence: $$a_{1}=2$$, $$a_{2}=0$$, $$a_{3}=3$$, $$a_{4}=1$$, and $$a_{5}=4$$

Key Concepts

Precalculus

When you're working with sequences, like the one from the exercise, you're diving into the world of precalculus. Precalculus is a branch of mathematics that prepares you for calculus by introducing concepts such as functions, sequences, and series. These concepts form the foundational language and tools you'll frequently use in calculus and beyond.
Understanding sequences is crucial in precalculus as they allow you to predict and analyze patterns. Recursively defined sequences, which we'll explore more here, are a special type of sequence where each term is defined based on its predecessors. This approach paves the way for understanding more complex mathematical ideas later on.

• Sequences are ordered lists of numbers.
• Each number is called a term.
• They teach you about mathematical patterns.

By mastering sequences in precalculus, you're setting yourself up for success in higher-level math and various real-world applications.

Sequence Definition

A sequence is essentially a set of numbers listed in a specific order. In many cases, you define a sequence using a formula that helps generate the sequence's terms. The specific type of sequence we see in the exercise is a recursive sequence.
A recursively defined sequence specifies the first term explicitly and then provides a rule for determining subsequent terms using previous terms. In the exercise, the sequence begins with \(a_1 = 2\) and uses the rule \(a_n = n - a_{n-1}\) for \(n \geq 2\). This method of defining sequences is powerful as it allows you to define complex patterns in a simple way.

• Begins with an initial term.
• Has a rule to find future terms.
• Relates each term to its previous terms.

Understanding the sequence definition is essential as it provides the tool necessary to identify any pattern and predict further numbers.

Term Finding

Finding the terms in a recursive sequence involves using the relation provided to calculate successive terms based on the previous ones. In the exercise, we find terms by using the recursive formula.
Starting with the known term \(a_1 = 2\), you calculate each following term by substituting back into the provided formula. Here’s how:

• For \(a_2\), use \(a_2 = 2 - a_1 = 0\).
• For \(a_3\), use \(a_3 = 3 - a_2 = 3\).
• Follow the process for \(a_4\) and \(a_5\).

These calculations demonstrate how to determine terms methodically by following the given recursive relationship, ensuring each term fits the pattern set by the sequence's definition. Understanding this process helps you decipher the sequence's structure.

Mathematical Induction

Mathematical induction is a powerful proof technique often used in mathematics to show that a statement holds true for all natural numbers. Even though the original exercise does not explicitly require proof by induction, understanding this concept can reinforce your confidence in reasoning about sequences.
Here's a basic outline of how mathematical induction works:

• Base Case: Verify the statement is true for the initial number, often \(n=1\).
• Inductive Step: Assume the statement holds for \(n=k\), and prove it also holds for \(n=k+1\).

While not used overtly in finding terms of a sequence in this exercise, knowing that sequences and their properties can often be proven using induction adds another layer to your mathematical toolbox. Proving the correctness or properties of recursive sequences can often hinge on this technique.