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Chapter 9: Problem 34

Express the sum in terms of summation notation. (Answers are not unique.) $$\frac{5}{13}+\frac{10}{11}+\frac{15}{9}+\frac{20}{7}$$

Short Answer

Expert verified
\( \sum_{n=1}^{4} \frac{5n}{15 - 2n} \)

Step by step solution

01

Identify the Pattern

Look at the numerators and denominators of each fraction to identify a pattern. In the numerators, we have 5, 10, 15, and 20, which can be expressed as multiples of 5 (i.e., 5n where n is an integer). For the denominators, 13, 11, 9, and 7 form a decreasing sequence subtracting 2 each time starting from 13.
02

Define the Sequence

We can express each term in the sequence using the general term \( \frac{5n}{(15 - 2n)} \). Here, for \( n = 1 \), the first term is \( \frac{5 \cdot 1}{15 - 2 \cdot 1} = \frac{5}{13} \), for \( n = 2 \), the second term is \( \frac{5 \cdot 2}{15 - 2 \cdot 2} = \frac{10}{11} \), and so forth.
03

Establish Limits for Summation

Determine the values for which n should run. Since there are four terms, and n starts at 1, it runs from 1 to 4 (inclusive) in our sequence expression.
04

Write in Summation Notation

Using the pattern from Steps 1-3, we can now write the entire sum as a single expression with summation notation: \[ \sum_{n=1}^{4} \frac{5n}{15 - 2n} \]. This compactly represents the entire summation operation.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Sequence Patterns
Sequence patterns play a crucial role in identifying how terms in a sequence relate to each other. When looking at a sequence, it鈥檚 essential to find what connects the terms. Recognizing a pattern helps in predicting subsequent terms and understanding the structure of the sequence.

To identify a sequence pattern:
  • Analyze how each term changes when compared to the previous term.
  • Look for arithmetic patterns, such as consistent addition or subtraction.
  • Check for geometric patterns, where terms are multiplied or divided by a constant.

In the exercise, the numerators follow an arithmetic pattern of multiples of 5, while the denominators form a sequence decreasing by 2 each time. Recognizing and expressing these patterns is the first step in forming a summation.
Series Representation
Series representation allows us to write the sum of a sequence in a more compact and elegant form using summation notation. When terms of a sequence are added together, they form a series.

The general process is:
  • Identify the pattern or rule of the sequence.
  • Express each term of the sequence using a general formula.
  • Use the formula to write the series in simplified summation form.

In the solution, the series \(\frac{5}{13}+\frac{10}{11}+\frac{15}{9}+\frac{20}{7}\)
is expressed as \(\sum_{n=1}^{4} \frac{5n}{15 - 2n}\), where each individual term\(\frac{5n}{(15 - 2n)}\)reflects the identified pattern of the sequence, making the series manageable and easy to interpret.
Mathematical Notation
Mathematical notation acts as a universal language that communicates complex mathematical thoughts with precision and clarity. Summation notation, in particular, is a powerful tool for representing series, especially those with a clear pattern.

Summation notation, represented by the symbol \( \sum \), encapsulates a series of terms expressed through a formula. Here's a quick guide:
  • The lower limit of the summation, beneath the \( \sum \), indicates where the summation begins.
  • The upper limit, above the \( \sum \), indicates where the summation ends.
  • Each term is described by a formula that uses a variable, often \( n \), which iterates from the lower to the upper limit.

In our example, \( \sum_{n=1}^{4} \frac{5n}{15 - 2n} \) concisely represents the sum of the terms in the defined sequence as \( n \) runs from 1 to 4. Understanding how to create and interpret this notation is key to mastering series representation.

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