Problem 9
Question
Write the first five terms of each sequence. Do not use a calculator. $$a_{n}=\frac{4 n-1}{n^{2}+2}$$
Step-by-Step Solution
Verified Answer
The first five terms are 1, \(\frac{7}{6}\), 1, \(\frac{5}{6}\), \(\frac{19}{27}\).
1Step 1: Identify the sequence formula
The formula given for the sequence is \(a_{n} = \frac{4n - 1}{n^2 + 2}\). To find the first five terms, we need to substitute the values \(n = 1, 2, 3, 4, 5\) into this formula.
2Step 2: Find the first term
Substitute \(n = 1\) into the formula: \(a_{1} = \frac{4(1) - 1}{1^2 + 2} = \frac{3}{3} = 1\).
3Step 3: Calculate the second term
Substitute \(n = 2\) into the formula: \(a_{2} = \frac{4(2) - 1}{2^2 + 2} = \frac{7}{6}\).
4Step 4: Determine the third term
Substitute \(n = 3\) into the formula: \(a_{3} = \frac{4(3) - 1}{3^2 + 2} = \frac{11}{11} = 1\).
5Step 5: Evaluate the fourth term
Substitute \(n = 4\) into the formula: \(a_{4} = \frac{4(4) - 1}{4^2 + 2} = \frac{15}{18} = \frac{5}{6}\).
6Step 6: Compute the fifth term
Substitute \(n = 5\) into the formula: \(a_{5} = \frac{4(5) - 1}{5^2 + 2} = \frac{19}{27}\).
Key Concepts
Sequence FormulaFirst Five TermsSubstitution MethodFraction Simplification
Sequence Formula
A sequence is a set of numbers arranged in a specific order, and understanding its formula is the first step to identifying its terms. The sequence formula given in this problem is \( a_n = \frac{4n - 1}{n^2 + 2} \). This formula is important because it tells us how to calculate any term in the sequence depending on the value of \( n \).
Each term in the sequence is found by substituting different values of \( n \) into the formula. The position of each term, such as the first term, second term, and so on, is determined by the integer \( n \).
Understanding a sequence formula allows students to predict future terms and comprehend the sequence's general behavior. This task specifically focuses on finding the first five terms by substitution.
Each term in the sequence is found by substituting different values of \( n \) into the formula. The position of each term, such as the first term, second term, and so on, is determined by the integer \( n \).
Understanding a sequence formula allows students to predict future terms and comprehend the sequence's general behavior. This task specifically focuses on finding the first five terms by substitution.
First Five Terms
Finding the first five terms of a sequence involves substituting the integers 1 through 5 into the sequence formula. For our formula \( a_n = \frac{4n - 1}{n^2 + 2} \), we label the terms as \( a_1, a_2, a_3, a_4, \) and \( a_5 \).
- For the first term, substitute \( n = 1 \). This gives us \( a_1 = \frac{3}{3} = 1 \).
- For the second term, using \( n = 2 \), we get \( a_2 = \frac{7}{6} \).
- The third term, with \( n = 3 \), results in \( a_3 = \frac{11}{11} = 1 \).
- For \( n = 4 \), the fourth term becomes \( a_4 = \frac{15}{18} = \frac{5}{6} \).
- Finally, substituting \( n = 5 \) gives the fifth term \( a_5 = \frac{19}{27} \).
Calculating these terms provides the initial portion of the sequence, essentially laying the foundation for understanding its pattern.
- For the first term, substitute \( n = 1 \). This gives us \( a_1 = \frac{3}{3} = 1 \).
- For the second term, using \( n = 2 \), we get \( a_2 = \frac{7}{6} \).
- The third term, with \( n = 3 \), results in \( a_3 = \frac{11}{11} = 1 \).
- For \( n = 4 \), the fourth term becomes \( a_4 = \frac{15}{18} = \frac{5}{6} \).
- Finally, substituting \( n = 5 \) gives the fifth term \( a_5 = \frac{19}{27} \).
Calculating these terms provides the initial portion of the sequence, essentially laying the foundation for understanding its pattern.
Substitution Method
The substitution method is a straightforward process used to find specific terms in a sequence. By inserting different values of \( n \) into the given sequence formula, we obtain the terms at those positions.
This method involves replacing \( n \) with numbers of interest—in this exercise, \( n = 1, 2, 3, 4, 5 \). Each substitution gives us a new term value, which reveals part of the sequence's structure.
The benefit of this method is that it allows students to isolate individual terms and understand how each one is calculated. It is crucial for developing an intuition about sequences, as each calculation step shows how changes in \( n \) affect the outcome.
This method involves replacing \( n \) with numbers of interest—in this exercise, \( n = 1, 2, 3, 4, 5 \). Each substitution gives us a new term value, which reveals part of the sequence's structure.
The benefit of this method is that it allows students to isolate individual terms and understand how each one is calculated. It is crucial for developing an intuition about sequences, as each calculation step shows how changes in \( n \) affect the outcome.
Fraction Simplification
Fraction simplification is a valuable skill in sequence term calculations, especially when the terms are initially expressed as fractions. Simplifying involves reducing the fraction to its simplest form where the numerator and denominator share no common factors other than 1.
In the given exercise, this process is applied to terms like \( \frac{3}{3} \) and \( \frac{15}{18} \). For \( \frac{3}{3} \), it simplifies directly to 1. For \( \frac{15}{18} \), dividing both the numerator and denominator by their greatest common divisor, which is 3, yields \( \frac{5}{6} \).
Simplifying fractions makes the terms of a sequence easier to interpret and compare. It also confirms whether sequences can sometimes appear different initially but are equivalent, like \( \frac{11}{11} \) simplifying to 1.
In the given exercise, this process is applied to terms like \( \frac{3}{3} \) and \( \frac{15}{18} \). For \( \frac{3}{3} \), it simplifies directly to 1. For \( \frac{15}{18} \), dividing both the numerator and denominator by their greatest common divisor, which is 3, yields \( \frac{5}{6} \).
Simplifying fractions makes the terms of a sequence easier to interpret and compare. It also confirms whether sequences can sometimes appear different initially but are equivalent, like \( \frac{11}{11} \) simplifying to 1.
Other exercises in this chapter
Problem 9
Find \(a_{5}\) and \(a_{n}\) for each geometric sequence. $$a_{4}=243, r=-3$$
View solution Problem 9
Use mathematical induction to prove each statement. Assume that \(n\) is a positive integer. $$3+3^{2}+3^{3}+\cdots+3^{n}=\frac{3\left(3^{n}-1\right)}{2}$$
View solution Problem 10
Write the first five terms of each arithmetic sequence. Do not use a calculator. $$a_{1}=4, d=3$$
View solution Problem 10
Find \(a_{5}\) and \(a_{n}\) for each geometric sequence. $$a_{4}=18, r=2$$
View solution