Problem 30 - Contemporary Precalculus

Question

Find the first five terms of the recursively defined sequence. $$a_{1}=-3 \text { and } a_{n}=(-1)^{n} 4 a_{n-1}-5 \quad \text { for } n \geq 2$$

Step-by-Step Solution

Answer

Question: Find the first five terms of the recursively defined sequence with the given conditions: \(a_1 = -3\) and \(a_n=(-1)^n 4a_{n-1}-5\) for \(n \geq 2\). Answer: The first five terms of the sequence are: -3, -17, 63, 247, and -993.

Step 1: Find the second term \(a_2\) using the recursive formula

Using the given formula, with \(n = 2\) and \(a_1 = -3\), we calculate: $$ a_2 = (-1)^2 4a_{1}-5 = (1)(4\cdot -3) - 5 = -12 - 5 = -17 $$ So, \(a_2 = -17\).

Step 2: Find the third term \(a_3\) using the recursive formula

Now, using the \(a_2\) value calculated above, we find \(a_3\), with \(n = 3\): $$ a_3 = (-1)^3 4a_{2}-5 = (-1)(4\cdot -17) - 5 = 68 - 5 = 63 $$ So, \(a_3 = 63\).

Step 3: Find the fourth term \(a_4\) using the recursive formula

Next, we find \(a_4\) using the \(a_3\) value, with \(n = 4\): $$ a_4 = (-1)^4 4a_{3}-5 = (1)(4\cdot 63) - 5 = 252 - 5 = 247 $$ So, \(a_4 = 247\).

Step 4: Find the fifth term \(a_5\) using the recursive formula

Finally, we find \(a_5\) using the \(a_4\) value, with \(n = 5\): $$ a_5 = (-1)^5 4a_{4}-5 = (-1)(4\cdot 247) - 5 = -988 - 5 = -993 $$ So, \(a_5 = -993\).

Step 5: List the first five terms of the sequence

By using the recursive formula, we have calculated the first five terms of the sequence as follows: $$ a_1 = -3, a_2 = -17, a_3 = 63, a_4 = 247, a_5 = -993 $$

Key Concepts

Sequence Calculation

Calculating a sequence involves finding each term based on a specific mathematical relationship, such as a formula or rule. In a sequence, each number is called a term, and the process of calculation may vary, ranging from using explicit formulas to recursive methods. For this particular sequence, we used recursive calculation. This term means using previous terms to determine the value of the next term. Each step builds on the last, leading to a sequence of values based solely on given initial information and the recursive formula we apply.

Imagine it as a chain reaction where one term links to another. You need to start from a known point, known as the base or initial term, like starting a domino effect—if the first one falls, the others follow based on their instruction.

Recursive Formula

A recursive formula is a specific kind of rule in mathematics to generate terms of a sequence where each term is expressed as a function of the preceding terms. For instance, in our sequence, the recursive formula given is \( a_{n} = (-1)^{n} 4 a_{n-1} - 5 \). Here, the value of each term depends on the previous term. We start from an initial term \( a_1 = -3 \) and use this formula to find all subsequent terms.

In essence, the recursive formula acts like a recipe guiding us on how to "cook up" the next term from the ingredients listed in the prior term. This method is particularly powerful because it builds out potentially complex sequences using simple steps. Notice how we needed to know the last term calculated to unlock the value of the current term.

Sequence Terms

Sequence terms are the individual elements within a sequence. In our example, each term is labeled as \(a_n\), where \(n\) denotes its position in the sequence. This labeling helps when working out and identifying patterns or implementing mathematical operations like recursive calculations.

Understanding how to calculate each term is key, as seen in the sequence: \(a_1 = -3\), \(a_2 = -17\), \(a_3 = 63\), \(a_4 = 247\), and \(a_5 = -993\). It is the recursive nature that determines each term based on the one that came before. As each calculation feeds into the next, learning how to compute these relies on a solid grasp of both the starting terms and the recursive operations applied.

Mathematical Sequences

Mathematical sequences comprise a set of numbers arranged in a particular order based on a defined rule or formula. They are fundamental concepts in mathematics, often serving as the basis for deeper studies in calculus and number theory. The order and relationship between terms are what make sequences fascinating and important for understanding mathematical patterns.

In the example given, the sequence starts with an initial term \(a_1 = -3\) and a defined pattern through a recursive formula. These sequences are instrumental in mathematical modeling across diverse applications, proving their versatility and necessity in varied fields of study. They demonstrate how a simple rule can unfold into an intricate set of numbers, revealing hidden relationships and helping solve complex problems.