Problem 61 - Beginning and Intermediate Algebra

Question

Write each series using summation notation. 3-9+27-81 \quad \sum_{i=1}^{4}(-1)^{i+1} \cdot\left(3^{i}\right)

Step-by-Step Solution

Answer

The given series 3 - 9 + 27 - 81 can be written using summation notation as follows: \[\sum_{i=1}^{4}(-1)^{i+1} \cdot 3^{i}\]

Step 1: Identify the pattern

We can see that the terms of the series are alternating between positive and negative numbers. We can represent this using the expression (-1)^{i+1}, where i is the term number. When i is even, the term will be positive, and when i is odd, the term will be negative.

Step 2: Identify the power of 3 in each term

In our series, we notice that the numbers are powers of 3. The power is equal to the term number i (i.e., 3^1, 3^2, 3^3, 3^4).

Step 3: Combine the alternating sign and power of 3 expressions

Now we combine the expressions for the alternating sign and the power of 3 into a single expression: (-1)^{i+1} * 3^{i}. Using this formula, we can generate terms for the series as follows: i = 1: (-1)^{1+1} * 3^{1} = 3 (first term) i = 2: (-1)^{2+1} * 3^{2} = -9 (second term) i = 3: (-1)^{3+1} * 3^{3} = 27 (third term) i = 4: (-1)^{4+1} * 3^{4} = -81 (fourth term)

Step 4: Write the series using summation notation

There are 4 terms in the series, so we write the summation notation as follows: \[\sum_{i=1}^{4}(-1)^{i+1} * 3^{i}\] This summation denotes the series 3 - 9 + 27 - 81, which is the given series.

Key Concepts

Alternating Series

In mathematics, an alternating series is a series where the signs of the terms alternate between positive and negative. An alternating series can be identified by its pattern of signs. For instance, in the given series (3 - 9 + 27 - 81), the terms switch signs from positive to negative as they increase.
This alternating sign pattern is conveniently represented with the term \((-1)^{i+1}\), where \(i\) is the term index. When \(i\) is odd, \((-1)^{i+1}\) results in a positive term, while an even \(i\) results in a negative term.
This mathematical trick of representing alternating series is crucial for understanding and expressing patterns concisely using summation, especially when dealing with complex series.

Powers of Three

The concept of powers of three in this series refers to the exponential increase of each term based on three as the base. This follows the pattern \(3^1, 3^2, 3^3,\) and so on. Powers of numbers are commonly used in sequences and have a significant role in the representations of series.
For the series 3, 9, 27, 81, each number can be expressed as a power of three:

• 3 is \(3^1\)
• 9 is \(3^2\)
• 27 is \(3^3\)
• 81 is \(3^4\)

The sequence grows rapidly, and recognizing this pattern can help in reducing complicated expressions into simpler, more recognizable forms in mathematics.

Series Representation

Representing series in a concise way is an essential skill in mathematics. It allows mathematicians and students alike to express long expressions using summation notation. The example given shows how an expression is derived by merging patterns within a series.
By combining the alternating series sign with powers of three, we engage both traits in a single mathematical expression: \((-1)^{i+1} \cdot 3^{i}\).
This representation simplifies communication and manipulation of mathematical ideas.
Understanding the series internally, one can represent complex patterns clearly and solve problems more effectively.

Summation Formula

Summation notation is an efficient way to sum a sequence of numbers that follow a recognizable pattern. It provides a framework to encapsulate a series in a compact formula. This is valuable, especially as the numbers in a sequence grow large or the sequence becomes elaborate.
The use of summation notation is shown as \(\sum_{i=1}^{n} a_i \), where each term is a function of \(i\), the index of summation.
For the series in question: \(\sum_{i=1}^{4}(-1)^{i+1} \cdot 3^{i}\), the notation conveys both the sign alternation and the powers of three in one gesture.
This structured form manages complexity, making analytics and calculations clearer and more manageable, both of which are key to higher-level mathematics.