Problem 4 - College Algebra

Question

What happens to the terms \(a_{n}\) of a sequence when there is a negative factor in the formula that is raised to a power that includes \(n\) ? What is the term used to describe this phenomenon?

Step-by-Step Solution

Answer

Alternating sequence.

Step 1: Understand the Sequence Formula

When a negative factor in the sequence formula is raised to a power including \(n\), it impacts the sign of the terms. For example, consider the sequence \(a_n = (-1)^n \cdot b_n\).

Step 2: Analyze the Pattern

Observe that when \(n\) is even, \((-1)^n\) is positive (since \((-1)^2 = 1\)), and when \(n\) is odd, \((-1)^n\) is negative (since \((-1)^1 = -1\)). This results in a sequence where the terms alternate in sign.

Step 3: Define the Term for This Phenomenon

The phenomenon of alternating signs in sequence terms due to a negative factor being raised to the power of \(n\) is called "alternating sequence."

Key Concepts

Sequence Formula

In mathematics, a sequence formula is an equation that describes the general term of a sequence. This formula allows us to calculate any term in the series given its position, denoted by the variable \( n \). Understanding the sequence formula is key to identifying the pattern or rule governing the sequence. For instance, the sequence formula \( a_n = (-1)^n \cdot b_n \) consists of two main components: the factor \((-1)^n\), which introduces an interesting behavior of alternating signs, and \(b_n\), which represents another sequence or rule that could be any well-defined series.

Negative Factor

In sequences like \( a_n = (-1)^n \cdot b_n \), a negative factor such as \((-1)\) plays a pivotal role. When this negative factor is raised to the power involving \( n \), it greatly influences the sequence by affecting the signs of its terms. The presence of a negative factor adds an alternating effect:

• When combined with even \( n \), the negative factor results in a positive product (since \((-1)^2 = 1\)).
• When paired with odd \( n \), it results in a negative product (since \((-1)^1 = -1\)).

This occurrence provides the foundation for alternating sequences, where each subsequent term flips its sign based on whether \( n \) is even or odd. The negative factor thus is integral in forming the signature alternating pattern.

Alternating Signs

The concept of alternating signs within a sequence directly stems from applying a negative factor raised to the power of \( n \), such as \( (-1)^n \). It simply means that each term in the sequence will have a positive sign followed by a negative sign and continue this pattern indefinitely:

• For \( n = 0, a_0 = (-1)^0 \cdot b_0 = 1 \cdot b_0 = b_0 \) is positive.
• For \( n = 1, a_1 = (-1)^1 \cdot b_1 = -1 \cdot b_1 = -b_1 \) is negative.
• The pattern repeats for higher values of \( n \).

This creation of alternating signs introduces a unique rhythmic structure into the sequence, making it very different from sequences where all terms have the same sign. This distinctive pattern is known as an "alternating sequence." It is commonly used in mathematical studies and applications, especially in series expansions and signal processing.

Power of n

The power of \( n \), represented as the exponent in mathematical sequences, determines how a factor, especially a negative one like \( (-1) \), will influence the sequence over its terms. The exponent \( n \) shapes the behavior of the entire sequence by dictating whether the negative base will be multiplied an even or odd number of times, hence affecting the term's sign. For example:

• If \( n \) is even, any negative base raised to this power turns positive.
• If \( n \) is odd, the base remains negative after the power is applied.

This cyclic alternation introduced by the power of \( n \) ensures that each sequence term alternates in sign, creating what is known as an alternating sequence. Understanding the impact of \( n \) helps reveal deeper insights into the sequence's behavior and properties, allowing for predictions and manipulations to serve various mathematical models and real-world applications.