Problem 6
Question
Determine the values of the sequence \(\left\\{a_{n}\right\\}\) for \(n=0,1,2, \ldots, 5\) $$ a_{n}=\frac{1}{\sqrt{n+1}} $$
Step-by-Step Solution
Verified Answer
The sequence values for \( a_n \) where \( n=0 \) to \( n=5 \) are 1, \( \frac{1}{\sqrt{2}} \), \( \frac{1}{\sqrt{3}} \), \( \frac{1}{2} \), \( \frac{1}{\sqrt{5}} \), and \( \frac{1}{\sqrt{6}} \).
1Step 1: Identify the given sequence formula
The sequence is defined as \( a_n = \frac{1}{\sqrt{n+1}} \). This formula will be used to calculate each term in the sequence by substituting the corresponding value of \( n \).
2Step 2: Calculate the value for \( n=0 \)
Substitute \( n=0 \) into the sequence formula: \[ a_0 = \frac{1}{\sqrt{0+1}} = \frac{1}{1} = 1 \] So, \( a_0 = 1 \).
3Step 3: Calculate the value for \( n=1 \)
Substitute \( n=1 \) into the sequence formula: \[ a_1 = \frac{1}{\sqrt{1+1}} = \frac{1}{\sqrt{2}} \] So, \( a_1 = \frac{1}{\sqrt{2}} \).
4Step 4: Calculate the value for \( n=2 \)
Substitute \( n=2 \) into the sequence formula: \[ a_2 = \frac{1}{\sqrt{2+1}} = \frac{1}{\sqrt{3}} \] So, \( a_2 = \frac{1}{\sqrt{3}} \).
5Step 5: Calculate the value for \( n=3 \)
Substitute \( n=3 \) into the sequence formula: \[ a_3 = \frac{1}{\sqrt{3+1}} = \frac{1}{\sqrt{4}} = \frac{1}{2} \] So, \( a_3 = \frac{1}{2} \).
6Step 6: Calculate the value for \( n=4 \)
Substitute \( n=4 \) into the sequence formula: \[ a_4 = \frac{1}{\sqrt{4+1}} = \frac{1}{\sqrt{5}} \] So, \( a_4 = \frac{1}{\sqrt{5}} \).
7Step 7: Calculate the value for \( n=5 \)
Substitute \( n=5 \) into the sequence formula: \[ a_5 = \frac{1}{\sqrt{5+1}} = \frac{1}{\sqrt{6}} \] So, \( a_5 = \frac{1}{\sqrt{6}} \).
Key Concepts
Understanding the Square Root FunctionTerm Calculation in SequencesExploring Mathematical Sequences
Understanding the Square Root Function
The square root function is a significant mathematical tool that helps simplify complex calculations. The function, denoted as \( \sqrt{x} \), determines a number which, when multiplied by itself, yields \( x \). For example, \( \sqrt{9} = 3 \) because \( 3 \times 3 = 9 \). The square root function is pivotal in determining the magnitude of numbers, especially those under a radical sign.
In the given sequence, \( a_n = \frac{1}{\sqrt{n+1}} \), the square root function assists in calculating the denominator of each term. Here, \( n \) is incremented step by step, determining the base of the square root. For instance, when \( n = 0 \), we get \( \sqrt{0 + 1} = \sqrt{1} = 1 \), simplifying the calculation of the sequence's terms.
In the given sequence, \( a_n = \frac{1}{\sqrt{n+1}} \), the square root function assists in calculating the denominator of each term. Here, \( n \) is incremented step by step, determining the base of the square root. For instance, when \( n = 0 \), we get \( \sqrt{0 + 1} = \sqrt{1} = 1 \), simplifying the calculation of the sequence's terms.
- Useful in handling quadratic equations
- Essential for defining geometric shapes and measurements
- Key in calculating distances and lengths in various applications
Term Calculation in Sequences
Calculating terms in a sequence involves substituting values into a given formula. For the sequence given by \( a_n = \frac{1}{\sqrt{n+1}} \), each term is discovered by replacing \( n \) with consecutive integers, starting with \( n = 0 \). You determine each term one at a time, providing a systematic approach to achieve the necessary results.
Here's how the process unfolds:
In the sequence calculation, you can observe each term, discover patterns, and gain a clearer understanding of the sequence's behavior. It's like following clear steps with a logical progression, making the whole process smooth and understandable.
Here's how the process unfolds:
- Substitute \( n = 0 \) in the formula: \( a_0 = \frac{1}{\sqrt{0+1}} = 1 \)
- Next, use \( n = 1 \): \( a_1 = \frac{1}{\sqrt{1+1}} = \frac{1}{\sqrt{2}} \)
- Repeat for \( n = 2 \), yielding \( a_2 = \frac{1}{\sqrt{3}} \)
- Continue this substitution up to \( n = 5 \)
In the sequence calculation, you can observe each term, discover patterns, and gain a clearer understanding of the sequence's behavior. It's like following clear steps with a logical progression, making the whole process smooth and understandable.
Exploring Mathematical Sequences
Mathematical sequences are ordered lists of numbers, usually defined by a specific rule or formula. They frequently appear in mathematics as they help structure and solve numerous mathematical problems. Sequences can be finite or infinite, depending on their rules and real-life applications.
The sequence \( a_n = \frac{1}{\sqrt{n+1}} \) is a simple example, forming a decreasing pattern as \( n \) increases. Each term finds its place by following the sequence formula, setting a concrete connection between consecutive terms. This pattern enables mathematicians and learners alike to predict subsequent values and understand deeper properties.
By studying sequences, you learn about patterns and their progression, which is essential in predicting outcomes, solving real-world problems, and further exploring mathematical theories.
The sequence \( a_n = \frac{1}{\sqrt{n+1}} \) is a simple example, forming a decreasing pattern as \( n \) increases. Each term finds its place by following the sequence formula, setting a concrete connection between consecutive terms. This pattern enables mathematicians and learners alike to predict subsequent values and understand deeper properties.
- Sequences are foundational in fields such as calculus and analysis
- They provide essential insights into patterns and behaviors in mathematics
- Crucial for understanding growth, decay, and limits in various contexts
By studying sequences, you learn about patterns and their progression, which is essential in predicting outcomes, solving real-world problems, and further exploring mathematical theories.
Other exercises in this chapter
Problem 6
Assume that the population growth is described by the Beverton-Holt recruitment curve with growth parameter \(R\) and carrying capacity \(K .\) For the given va
View solution Problem 6
give a formula for \(N(t), t=0,1,2, \ldots\), on the basis of the information provided. \(N_{0}=4\); population doubles every 40 minutes; one unit of time is 40
View solution Problem 7
give a formula for \(N(t), t=0,1,2, \ldots\), on the basis of the information provided. \(N_{0}=1\); population doubles every 40 minutes; one unit of time is 80
View solution Problem 7
Determine the values of the sequence \(\left\\{a_{n}\right\\}\) for \(n=0,1,2, \ldots, 5\) $$ f(n)=(n+1)^{2} $$
View solution