Problem 28 - Algebra and Trigonometry with Analytic Geometry

Question

Express the sum in terms of summation notation. (Answers are not unique.) $$2-4+8-16+32-64$$

Step-by-Step Solution

Answer

\(\sum_{n=1}^{6} (-1)^{n+1} \cdot 2^n\)

Step 1: Analyze the Pattern

First, we observe the given series: \(2, -4, 8, -16, 32, -64\). Notice the signs alternate as positive and negative, and each term is a power of 2.

Step 2: Find the General Term

Identify the expression for the \(n\)-th term of the series. The absolute value of each term is \(2^n\), starting from \(n=1\). To account for the alternating signs, the general term can be expressed as \((-1)^{n+1} \cdot 2^n\).

Step 3: Write in Summation Notation

Now, write the series using summation notation. Since there are 6 terms, the series can be expressed as: \[ \sum_{n=1}^{6} (-1)^{n+1} \cdot 2^n \]

Step 4: Verify the Expression

Verify the summation expression by calculating a few initial terms: for \(n=1\), the term is \(2\); for \(n=2\), the term is \(-4\), and so on. This matches the original series.

Key Concepts

Series Patterns

In mathematics, series patterns are fascinating because they reveal how sequences of numbers behave over time. A series is essentially the sum of elements of a sequence. In the given series, we notice a distinct pattern: each term increases in magnitude, and the signs alternate between positive and negative. Recognizing such patterns can help us write and analyze the series efficiently.

• The series comprises powers of a base number, which is 2 in this case. Each subsequent term is the previous term multiplied by 2.
• The alternating signs can be a clue to the use of powers of -1.

Understanding the pattern is crucial because it enables the efficient use of mathematical tools, such as summation notation, to express complex series in a compact form.

Alternating Series

An alternating series is a series in which the sign of each term switches back and forth. This is a common occurrence in mathematical series and can often be spotted through its pattern of positive and negative terms. In our series, this concept is clearly illustrated:

• Alternating series are often expressed using \((-1)^n\).
• The expression \((-1)^{n+1}\) ensures the correct starting sign. It is positive for odd-numbered terms when n starts from 1.

When we use summation notation to handle alternating series, it saves time and makes it easier to verify calculations. Alternating series are essential in various mathematical applications, including physics and engineering, where oscillations often occur.

Exponential Functions

Exponential functions are mathematical functions involving exponents. In this series, each term can be represented as a power of 2, demonstrating exponential growth. When analyzing exponential functions within a series:

• Each term in the sequence, such as \(2^n\), shows how quickly numbers can grow with a constant base and varying exponent.
• This growth pattern is parallel to compounded interest or population growth scenarios.

Recognizing exponential functions within a series can provide insights into how rapidly things increase or decrease. It's an important concept in both real-world applications and theoretical mathematics.

General Term

The general term of a sequence or series is an expression that allows you to find any term in the series without having to write out all preceding terms. In the provided series, the general term is \((-1)^{n+1} \cdot 2^n\). Understanding the general term:

• The factor \((-1)^{n+1}\) controls the alternating sign, which is essential for capturing the behavior of alternating series.
• The \(2^n\) part captures the exponential growth of the terms.
• Being able to determine the general term is key in using summation notation, as it is compact and simplifies the expression of series.

Identifying the general term is crucial in moving from a list of numbers to a mathematical expression that can describe an entire series.