Problem 70
Question
Write the series with summation notation. Let the lower limit equal 1. $$1+\frac{1}{5}+\frac{1}{25}+\frac{1}{125}+\frac{1}{625}$$
Step-by-Step Solution
Verified Answer
\(\sum_{k=1}^{5} \frac{1}{5^{(k-1)}}\)
1Step 1: Identify the Pattern
Examine the series provided: \(1 + \frac{1}{5} + \frac{1}{25} + \frac{1}{125} + \frac{1}{625}\). Notice that each term after the first is of the form \(\frac{1}{5^n}\), where \(n\) is an increasing integer starting from 0.
2Step 2: Write the General Term
The general term of the series can be expressed as \(\frac{1}{5^{(k-1)}}\), where \(k\) is the term number. This accounts for starting from 1 when \(k = 1\).
3Step 3: Determine the Number of Terms
Count the number of terms in the series. The series provided has 5 terms: \(1, \frac{1}{5}, \frac{1}{25}, \frac{1}{125}, \frac{1}{625}\).
4Step 4: Write in Summation Notation
Use the summation symbol to express the series: \(\sum_{k=1}^{5} \frac{1}{5^{(k-1)}}\). This represents the sum of the first 5 terms of the described sequence with the initial term corresponding to \(k=1\).
Key Concepts
Series RepresentationGeometric SequenceAlgebraic Patterns
Series Representation
When dealing with mathematical series, it's crucial to know how to represent them in a concise form. This is where summation notation, also known as sigma notation, becomes very helpful. Summation notation uses the Greek letter sigma (\(\Sigma\)) to represent the summation of a sequence of terms. Each element in the series is the result of a general formula based on the term's position in the series.
This notation helps simplify complex series by expressing them in a compact form, clarifying both the structure and the number of terms to be summed. For instance, the series given: \(1 + \frac{1}{5} + \frac{1}{25} + \frac{1}{125} + \frac{1}{625}\), can be represented in summation notation as \(\sum_{k=1}^{5} \frac{1}{5^{(k-1)}}\). This notation highlights the pattern in the terms, making it easier to analyze and understand the series structure.
This notation helps simplify complex series by expressing them in a compact form, clarifying both the structure and the number of terms to be summed. For instance, the series given: \(1 + \frac{1}{5} + \frac{1}{25} + \frac{1}{125} + \frac{1}{625}\), can be represented in summation notation as \(\sum_{k=1}^{5} \frac{1}{5^{(k-1)}}\). This notation highlights the pattern in the terms, making it easier to analyze and understand the series structure.
Geometric Sequence
A geometric sequence is a series of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. In the series provided, the common ratio \(r\) is \(\frac{1}{5}\). This means each term is obtained by multiplying the previous term by the fraction \(\frac{1}{5}\).
Geometric sequences are common in mathematics because of their straightforward pattern and growth. Understanding the concept of a geometric sequence is crucial for tackling series problems, as it allows for the prediction and computation of any term in the series using only the first term and the common ratio. For this series, the first term is \(1\), and each subsequent term is derived by multiplying the previous term by \(\frac{1}{5}\).
To visualize this, consider the sequence starting from \(1\):
Geometric sequences are common in mathematics because of their straightforward pattern and growth. Understanding the concept of a geometric sequence is crucial for tackling series problems, as it allows for the prediction and computation of any term in the series using only the first term and the common ratio. For this series, the first term is \(1\), and each subsequent term is derived by multiplying the previous term by \(\frac{1}{5}\).
To visualize this, consider the sequence starting from \(1\):
- 1st term: \(1\)
- 2nd term: \(1 \times \frac{1}{5} = \frac{1}{5}\)
- 3rd term: \(\frac{1}{5} \times \frac{1}{5} = \frac{1}{25}\)
- 4th term: \(\frac{1}{25} \times \frac{1}{5} = \frac{1}{125}\)
- 5th term: \(\frac{1}{125} \times \frac{1}{5} = \frac{1}{625}\)
Algebraic Patterns
An algebraic pattern is a sequence of numbers, expressions, or objects that follow a specific rule or formula. Recognizing these patterns allows students to write formulas that define the entire sequence. These can often be identified and defined using mathematical operations or relations.
In the provided series, the algebraic pattern arises from the way each term is generated: specifically, we see a power pattern. Each term can be expressed as \(\frac{1}{5^{(k-1)}}\), derived from exponentiation where the base is consistent (\(5\) in this case), and the exponent is dependent on the term index \(k\).
Identifying algebraic patterns in sequences enables the prediction of future terms and the understanding of the sequence’s behavior. It simplifies complex series into recognizable general forms, essential for further mathematical processing like calculating sums in long sequences using formulae rather than listing all terms.
In the provided series, the algebraic pattern arises from the way each term is generated: specifically, we see a power pattern. Each term can be expressed as \(\frac{1}{5^{(k-1)}}\), derived from exponentiation where the base is consistent (\(5\) in this case), and the exponent is dependent on the term index \(k\).
Identifying algebraic patterns in sequences enables the prediction of future terms and the understanding of the sequence’s behavior. It simplifies complex series into recognizable general forms, essential for further mathematical processing like calculating sums in long sequences using formulae rather than listing all terms.
- Recognize how exponents are changing in each step.
- Identify base numbers to form a general pattern.
- Express such patterns in a mathematical formula for ease of calculation.
Other exercises in this chapter
Problem 70
Determine if \(f\) is a geometric sequence. $$f(n)=-3(0.25)^{n}$$
View solution Problem 70
On a test with 6 essay questions, students are asked to answer 4 questions. How many ways can the essay questions be selected? in the second part a student must
View solution Problem 71
Write the series with summation notation. Let the lower limit equal 1. $$1+\frac{4}{3}+\frac{6}{4}+\frac{8}{5}+\frac{10}{6}+\frac{12}{7}+\frac{14}{8}$$
View solution Problem 72
Determine if \(f\) is a geometric sequence. $$f(n)=2(n-1)^{n}$$
View solution