Problem 16 - College Algebra

Question

For the following exercises, write the first eight terms of the piecewise sequence. $$ a_{n}=\left\\{\begin{array}{l} (-2)^{n}-2 \text { if } n \text { is even } \\ (3)^{n-1} \text { if } n \text { is odd } \end{array}\right. $$

Step-by-Step Solution

Answer

1, 2, 9, 14, 81, 62, 729, 254.

Step 1: Identify the Sequence Rule

The sequence is defined by a piecewise function. For even indices \( n \), the term is given by \( (-2)^{n} - 2 \), and for odd indices \( n \), the term is given by \( (3)^{n-1} \). Our task is to find the first eight terms of this sequence using these rules.

Step 2: Calculate Term 1

Since \( n = 1 \) is odd, use the odd sequence rule \( (3)^{n-1} \). For \( n = 1 \), the term is \( a_1 = (3)^{1-1} = 3^0 = 1 \).

Step 3: Calculate Term 2

Since \( n = 2 \) is even, use the even sequence rule \( (-2)^n - 2 \). For \( n = 2 \), the term is \( a_2 = (-2)^2 - 2 = 4 - 2 = 2 \).

Step 4: Calculate Term 3

Since \( n = 3 \) is odd, use the odd sequence rule \( (3)^{n-1} \). For \( n = 3 \), the term is \( a_3 = (3)^{3-1} = 3^2 = 9 \).

Step 5: Calculate Term 4

Since \( n = 4 \) is even, use the even sequence rule \( (-2)^n - 2 \). For \( n = 4 \), the term is \( a_4 = (-2)^4 - 2 = 16 - 2 = 14 \).

Step 6: Calculate Term 5

Since \( n = 5 \) is odd, use the odd sequence rule \( (3)^{n-1} \). For \( n = 5 \), the term is \( a_5 = (3)^{5-1} = 3^4 = 81 \).

Step 7: Calculate Term 6

Since \( n = 6 \) is even, use the even sequence rule \( (-2)^n - 2 \). For \( n = 6 \), the term is \( a_6 = (-2)^6 - 2 = 64 - 2 = 62 \).

Step 8: Calculate Term 7

Since \( n = 7 \) is odd, use the odd sequence rule \( (3)^{n-1} \). For \( n = 7 \), the term is \( a_7 = (3)^{7-1} = 3^6 = 729 \).

Step 9: Calculate Term 8

Since \( n = 8 \) is even, use the even sequence rule \( (-2)^n - 2 \). For \( n = 8 \), the term is \( a_8 = (-2)^8 - 2 = 256 - 2 = 254 \).

Step 10: Write the Sequence

The first eight terms of the sequence are: 1, 2, 9, 14, 81, 62, 729, 254.

Key Concepts

Odd and Even Functions

Understanding whether a function is odd or even is crucial when dealing with piecewise sequences. In mathematics, an even function is one where \( f(x) = f(-x) \) for all \( x \). This means the graph of the function is symmetric with respect to the y-axis. Odd functions, on the other hand, follow the rule \( f(-x) = -f(x) \), showing symmetry about the origin.
For example, the sequence given like \( a_{n} \) where terms differ based on whether the index \( n \) is odd or even, highlights the structural beauty of alternating mathematical behavior.

• **Even Indexed Terms**: These are calculated using the function \( (-2)^n - 2 \).
• **Odd Indexed Terms**: Are derived using the formula \( (3)^{n-1} \).

This separation allows a systematic computation of each sequence term, by checking the odd or even nature of \( n \). Understanding these concepts aids in grasping how functions behave and how it influences sequence outcomes.

Sequence Rules

Sequence rules are fundamental in defining the relationship between sequence terms. In the provided problem, the piecewise function lays out two distinct rules depending on the parity of \( n \).
Rules bring an order to sequences and help predict future terms:

• For \( n \) being even, the rule \( (-2)^n - 2 \) ensures that the sequence term is always reduced by 2 from the power of \(-2\).
• For \( n \) being odd, the sequence uses the rule \( (3)^{n-1} \), illustrating exponential growth based on powers of 3.

Thus, these sequence rules ensure each term can be systematically calculated and convey essential properties such as monotonicity or periodicity under different scenarios.

Algebraic Sequences

Algebraic sequences use mathematical expressions to define their terms, often seen in piecewise functions, as in this exercise. Algebraic sequences are versatile and can model various real-world problems.
The given sequence \( a_{n} \) is a combination of two algebraic expressions:

• The expression \( (-2)^n - 2 \) outlines a sequence where each even-indexed term decreases slightly from a high power value.
• Conversely, \( (3)^{n-1} \) governs the odd-indexed terms, showing rapid growth with increasing powers of three.

These expressions help illustrate how exponential and polynomial forms can drive sequence behaviors and outcomes, making algebraic sequences a valuable tool in mathematics.