Problem 4
Question
.List the first five terms of the sequence. \( a_n = \frac {n^2 - 1}{n^2 + 1} \)
Step-by-Step Solution
Verified Answer
0, \(\frac{3}{5}\), \(\frac{4}{5}\), \(\frac{15}{17}\), \(\frac{12}{13}\).
1Step 1: Identify the sequence formula
We are given the formula for the sequence as \( a_n = \frac{n^2 - 1}{n^2 + 1} \). To find the first five terms, we need to plug in the values of \( n \) from 1 to 5 into this formula.
2Step 2: Compute the first term
Substitute \( n = 1 \) into the sequence formula: \[a_1 = \frac{1^2 - 1}{1^2 + 1} = \frac{0}{2} = 0\]Therefore, the first term is 0.
3Step 3: Compute the second term
Substitute \( n = 2 \) into the sequence formula: \[a_2 = \frac{2^2 - 1}{2^2 + 1} = \frac{4 - 1}{4 + 1} = \frac{3}{5}\]Thus, the second term is \( \frac{3}{5} \).
4Step 4: Compute the third term
Substitute \( n = 3 \) into the sequence formula: \[a_3 = \frac{3^2 - 1}{3^2 + 1} = \frac{9 - 1}{9 + 1} = \frac{8}{10} = \frac{4}{5}\]So, the third term is \( \frac{4}{5} \).
5Step 5: Compute the fourth term
Substitute \( n = 4 \) into the sequence formula: \[a_4 = \frac{4^2 - 1}{4^2 + 1} = \frac{16 - 1}{16 + 1} = \frac{15}{17}\]Therefore, the fourth term is \( \frac{15}{17} \).
6Step 6: Compute the fifth term
Substitute \( n = 5 \) into the sequence formula: \[a_5 = \frac{5^2 - 1}{5^2 + 1} = \frac{25 - 1}{25 + 1} = \frac{24}{26} = \frac{12}{13}\]Thus, the fifth term is \( \frac{12}{13} \).
Key Concepts
Sequence FormulaTerms of a SequenceStep-by-Step Solution
Sequence Formula
Understanding a sequence begins with knowing its formula. A sequence formula allows us to generate the terms of the sequence by substituting different values of a variable, typically denoted by \( n \).
The most critical part about the sequence formula is learning how it systematically defines the pattern the sequence follows. In our case, the sequence formula given is:
\[ a_n = \frac{n^2 - 1}{n^2 + 1} \].
This formula tells us how to calculate any term \( a_n \) of the sequence based on the position \( n \) within that sequence.
By using this general formula, you can plug in any integer value for \( n \) to find the corresponding term in the sequence. It systematically ensures that each term is the result of a consistent and repeatable operation.
When deciphering sequence formulas, the first step usually involves recognizing this formula and pinpointing how it layers in operations—be it through exponentiation, addition, subtraction, and more—to define the sequence.
The most critical part about the sequence formula is learning how it systematically defines the pattern the sequence follows. In our case, the sequence formula given is:
\[ a_n = \frac{n^2 - 1}{n^2 + 1} \].
This formula tells us how to calculate any term \( a_n \) of the sequence based on the position \( n \) within that sequence.
By using this general formula, you can plug in any integer value for \( n \) to find the corresponding term in the sequence. It systematically ensures that each term is the result of a consistent and repeatable operation.
When deciphering sequence formulas, the first step usually involves recognizing this formula and pinpointing how it layers in operations—be it through exponentiation, addition, subtraction, and more—to define the sequence.
Terms of a Sequence
The terms of a sequence are the individual elements or numbers produced by a sequence formula. Each term corresponds to a unique position within the sequence, beginning with \( n = 1 \) for the first term.
To find the first few terms, of our sequence, we substitute consecutive integers into the sequence formula. Thus, for every position \( n \), we calculate \( a_n \). Here's how it looks like for the initial five terms:
To find the first few terms, of our sequence, we substitute consecutive integers into the sequence formula. Thus, for every position \( n \), we calculate \( a_n \). Here's how it looks like for the initial five terms:
- First term: Substitute \( n = 1 \) into the formula to find \( a_1 = \frac{1^2 - 1}{1^2 + 1} \). This calculates to 0.
- Second term: Substitute \( n = 2 \) to find \( a_2 = \frac{4 - 1}{4 + 1} = \frac{3}{5} \).
- Third term: Substitute \( n = 3 \) resulting in \( a_3 = \frac{9 - 1}{9 + 1} = \frac{4}{5} \).
- Fourth term: Substitute \( n = 4 \) for \( a_4 = \frac{16 - 1}{16 + 1} = \frac{15}{17} \).
- Fifth term: Substitute \( n = 5 \) to find \( a_5 = \frac{25 - 1}{25 + 1} = \frac{12}{13} \).
Step-by-Step Solution
Solving sequence problems like this one often requires a step-by-step approach, ensuring accuracy and clarity in finding the terms.
To start, identify and write down the given sequence formula.
This systematic approach ensures that each term is calculated correctly by adhering closely to the structure provided by the sequence's formula.
To start, identify and write down the given sequence formula.
- First, plug in \( n = 1 \) to compute the first term. Solve the operation to find the term \( a_1 \).
- Next, replace \( n \) with 2 to find the second term \( a_2 \) and perform the calculation.
- This approach is repeated for \( n = 3, 4, 5 \) to obtain the subsequent terms respectively.
This systematic approach ensures that each term is calculated correctly by adhering closely to the structure provided by the sequence's formula.
Other exercises in this chapter
Problem 4
Use the Integral Test to determine whether the series is convergent or divergent. \( \displaystyle \sum_{n = 1}^{\infty} n^{-0.3} \)
View solution Problem 4
Calculate the sum of the series \( \sum_{n = 1}^{\infty} a_n \) whose partial sums are given. \( s_n = \frac {n^2 - 1}{4n^2 + 1} \)
View solution Problem 5
Find the Taylor polynomials \( T_3(x) \) for the function \( f \) centered at the number \( a \) Graph \( f \) and \( T_3 \) on the same screen. \( f(x) = \cos
View solution Problem 5
Use the definition of a Taylor series to find the first four nonzero terms of the series for \( f(x) \) centered at the given value of \( a. \) \( f(x) = xe^x,
View solution