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Magazine Chapter 15: Problem 18
Given the general term of each sequence, find each of the following. \(a_{n}=\frac{3 n-1}{4 n+5}\) a) \(a_{1}\) b) \(a_{10}\) c) the 21 st term
Short Answer
Expert verified
The first term (\(a_1\)) is \(\frac{2}{9}\), the tenth term (\(a_{10}\)) is \(\frac{29}{45}\), and the 21st term is \(\frac{62}{89}\).
Step by step solution
01
Calculate the first term \(a_1\)
To find the first term of the sequence, we'll set n = 1 and substitute this value into the general term formula. So, we'll have \(a_1=\frac{3(1)-1}{4(1)+5}\).
02
Simplify the expression for \(a_1\)
Plugging the value of n = 1 into the formula, we get:
\(a_1=\frac{3(1)-1}{4(1)+5} = \frac{3-1}{4+5} = \frac{2}{9}\)
03
Calculate the tenth term \(a_{10}\)
To find the tenth term of the sequence, we'll set n = 10 and substitute this value into the general term formula. So, we'll have \(a_{10}=\frac{3(10)-1}{4(10)+5}\).
04
Simplify the expression for \(a_{10}\)
Plugging the value of n = 10 into the formula, we get:
\(a_{10}=\frac{3(10)-1}{4(10)+5} = \frac{30-1}{40+5} = \frac{29}{45}\)
05
Calculate the 21st term of the sequence
To find the 21st term of the sequence, we'll set n = 21 and substitute this value into the general term formula. So, we'll have \(a_{21}=\frac{3(21)-1}{4(21)+5}\).
06
Simplify the expression for the 21st term
Plugging the value of n = 21 into the formula, we get:
\(a_{21}=\frac{3(21)-1}{4(21)+5} = \frac{63-1}{84+5} = \frac{62}{89}\)
The computed terms of the sequence are as follows:
a) The first term (\(a_1\)) is \(\frac{2}{9}\)
b) The tenth term (\(a_{10}\)) is \(\frac{29}{45}\)
c) The 21st term is \(\frac{62}{89}\)
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
General Term of a Sequence
The general term of a sequence provides a formula you can use to find any term in that sequence. It is usually expressed in terms of \( n \), which represents the position of a term within the sequence. For example, the sequence described by the general term \( a_n = \frac{3n-1}{4n+5} \) allows you to find any term, whether it is the first term or the hundredth.
By substituting different values for \( n \), you can observe how the sequence behaves and identify specific numbers within the series.
This formula is critical in sequence analysis because it offers a way to describe an infinite list of numbers using just one mathematical expression.
By substituting different values for \( n \), you can observe how the sequence behaves and identify specific numbers within the series.
This formula is critical in sequence analysis because it offers a way to describe an infinite list of numbers using just one mathematical expression.
- Provides a method to determine each term in a sequence.
- Can be applied by substituting \( n \) with any natural number.
- Helps in identifying patterns and regularities within a sequence.
Finding Terms of a Sequence
To determine specific terms of a sequence, you'll need to substitute the position number of the term into the general formula. Let's go through the steps:
1. **Locate the position:** Identify which term you want, such as \( a_1 \), \( a_{10} \), or \( a_{21} \). This tells you the value of \( n \) you will use.
2. **Substitute and solve:** Replace \( n \) in the general term formula with the desired position number. For instance, replacing \( n = 1 \) in our example yields \( a_1 = \frac{3(1)-1}{4(1)+5} = \frac{2}{9} \).
3. **Simplify:** Perform any necessary arithmetic to get the simplest form of the term.
This process allows you to pinpoint exact terms within a sequence quickly.
1. **Locate the position:** Identify which term you want, such as \( a_1 \), \( a_{10} \), or \( a_{21} \). This tells you the value of \( n \) you will use.
2. **Substitute and solve:** Replace \( n \) in the general term formula with the desired position number. For instance, replacing \( n = 1 \) in our example yields \( a_1 = \frac{3(1)-1}{4(1)+5} = \frac{2}{9} \).
3. **Simplify:** Perform any necessary arithmetic to get the simplest form of the term.
This process allows you to pinpoint exact terms within a sequence quickly.
- Aids in calculating both the initial and longer-stage terms.
- Simplifies understanding how a sequence progresses from one term to the next.
Arithmetic Sequences
Arithmetic sequences are a special type of mathematical sequence where each successive term is derived by adding or subtracting a constant known as the common difference. For instance, in the sequence \(2, 4, 6, 8, \dots\), the common difference is \(2\).
While our example \( a_n = \frac{3n-1}{4n+5} \) is not an arithmetic sequence, understanding arithmetic sequences can help grasp different types of sequences.
Facts about arithmetic sequences:
While our example \( a_n = \frac{3n-1}{4n+5} \) is not an arithmetic sequence, understanding arithmetic sequences can help grasp different types of sequences.
Facts about arithmetic sequences:
- Each term increases or decreases by the same amount.
- Finding terms is straightforward once you know the common difference.
- General formula: \( a_n = a_1 + (n-1)d \) where \( d \) is the common difference.
Mathematical Sequences
Mathematical sequences can be classified into different types aside from arithmetic, such as geometric or even the more complex Fibonacci sequences. Each sequence type follows a unique rule for generating subsequent terms.
The general term \( a_n = \frac{3n-1}{4n+5} \) is an example of a mathematical sequence where terms are derived using a fraction involving \( n \). Different types include:
The general term \( a_n = \frac{3n-1}{4n+5} \) is an example of a mathematical sequence where terms are derived using a fraction involving \( n \). Different types include:
- **Geometric Sequences:** Multiply the previous term by a constant ratio. Formula: \( a_n = a_1 \times r^{(n-1)} \).
- **Fibonacci Sequence:** Each term is the sum of the two preceding ones, starting from 0 and 1.
