Problem 65 - Algebra and Trigonometry Real Mathematics, Real People

Question

Using Summation Notation Use summation notation to write the sum. $$5+15+45+\cdots+3645$$

Step-by-Step Solution

Answer

The sum of the series \(5+15+45+...+3645\) in summation (sigma) notation is \(\Sigma_{n=1}^{8} 5 \times 3^{n-1}\)

Step 1: Identify the pattern

Examining the series, it can be noticed that each term is three times the term before it. All terms can be represented using the formula \(5 \times 3^{n-1}\) where n starts at 1 and each subsequent term increments n by 1.

Step 2: Find the number of terms

With the series starting at 5 and ending at 3645, and since each term is thrice the previous term, there are 8 terms in total.

Step 3: Write in Sigma notation

The sigma notation of the series \(5+15+45+...+3645\) is \(\Sigma_{n=1}^{8} 5 \times 3^{n-1}\)

Key Concepts

Geometric Series

A geometric series is a sum of terms that have a constant ratio between consecutive terms. This ratio is known as the common ratio. For the series in the problem, each term is obtained by multiplying the previous term by 3. This identifies the series as a geometric progression with a common ratio of 3.
Key points to understand about geometric series include:

• The first term is the starting point, which in this series is 5.
• The common ratio, which tells us how each term relates to its predecessor, is 3 in this case.
• The formula for the n-th term is generally given by \(a \cdot r^{(n-1)}\), where \(a\) is the first term and \(r\) is the common ratio.

Geometric series are fundamental in mathematics due to their simple yet powerful structure, making them a key tool for solving many types of problems involving exponential growth or decay.
Understanding these concepts helps in identifying series patterns and efficiently solving related problems.

Sigma Notation

Sigma notation is a compact and powerful way to express the sum of a sequence of terms. It is denoted by the Greek letter \(\Sigma\), which stands for "sum."
In the problem, the series \(5+15+45+\cdots+3645\) is expressed using sigma notation as \(\Sigma_{n=1}^{8} 5 \times 3^{n-1}\). Here's what each part represents:

• The \(n=1\) below the \(\Sigma\) symbol signifies the starting index for the series.
• The superscript 8 above the \(\Sigma\) symbol is the ending index, indicating there are 8 total terms.
• The expression \(5 \times 3^{n-1}\) is the formula for each term in the series, describing how each term is calculated based on \(n\).

Sigma notation makes it easier to handle and manipulate large sums, especially when associated with geometric series or other patterns. It's commonly used in calculus and higher mathematics to deal with infinite series or sums of functions over intervals.

Series Representation

The representation of a series involves expressing it in a form that is concise yet informative, allowing one to easily grasp the nature of the sequence.
For example, in the original exercise, the series \(5 + 15 + 45 + \cdots + 3645\) is represented concisely in the step-by-step solution using sigma notation. This is important for several reasons:

• It provides a clear view of the terms involved and the pattern they follow.
• Facilitates the computation of the sum by providing a direct formula for each term.
• Simplifies the process of deducing properties of the series, such as the number of terms and the progression pattern.

Representing series efficiently is crucial for simplifying complex expressions and performing mathematical operations on them. It serves as an essential skill in mathematical problem-solving, making the process more manageable and intuitive for learners.