Problem 10
Question
Write the first five terms of each sequence. Do not use a calculator. $$a_{n}=\frac{n^{2}-1}{n^{2}+1}$$
Step-by-Step Solution
Verified Answer
The first five terms are 0, \( \frac{3}{5} \), \( \frac{4}{5} \), \( \frac{15}{17} \), \( \frac{12}{13} \).
1Step 1: Understand the given formula
The sequence is defined by the formula \( a_n = \frac{n^2 - 1}{n^2 + 1} \). Our task is to calculate the first five terms of this sequence by substituting the values of \( n \) from 1 to 5 into the given formula.
2Step 2: Calculate the first term \( a_1 \)
Substitute \( n = 1 \) into the formula: \\[ a_1 = \frac{1^2 - 1}{1^2 + 1} = \frac{0}{2} = 0 \] Thus, the first term is 0.
3Step 3: Calculate the second term \( a_2 \)
Substitute \( n = 2 \) into the formula: \\[ a_2 = \frac{2^2 - 1}{2^2 + 1} = \frac{4 - 1}{4 + 1} = \frac{3}{5} \] So, the second term is \( \frac{3}{5} \).
4Step 4: Calculate the third term \( a_3 \)
Substitute \( n = 3 \) into the formula: \\[ a_3 = \frac{3^2 - 1}{3^2 + 1} = \frac{9 - 1}{9 + 1} = \frac{8}{10} = \frac{4}{5} \] The third term is \( \frac{4}{5} \).
5Step 5: Calculate the fourth term \( a_4 \)
Substitute \( n = 4 \) into the formula: \\[ a_4 = \frac{4^2 - 1}{4^2 + 1} = \frac{16 - 1}{16 + 1} = \frac{15}{17} \] Hence, the fourth term is \( \frac{15}{17} \).
6Step 6: Calculate the fifth term \( a_5 \)
Substitute \( n = 5 \) into the formula: \\[ a_5 = \frac{5^2 - 1}{5^2 + 1} = \frac{25 - 1}{25 + 1} = \frac{24}{26} = \frac{12}{13} \] Therefore, the fifth term is \( \frac{12}{13} \).
7Step 7: Collect all first five terms
The first five terms are: 0, \( \frac{3}{5} \), \( \frac{4}{5} \), \( \frac{15}{17} \), \( \frac{12}{13} \).
Key Concepts
Understanding Algebraic ExpressionsSimplifying Fractions in SequencesCalculating Terms in a SequenceUsing Mathematical Substitution
Understanding Algebraic Expressions
An algebraic expression is a combination of numbers, variables, and operators (like plus or minus). In mathematics, expressions are used to represent numbers and define patterns in sequences.
In the given exercise, the algebraic expression is \( a_n = \frac{n^2 - 1}{n^2 + 1} \). Here, \( n \) is a variable representing the position of a term in the sequence.
As \( n \) changes, the expression evaluates to different values.
In the given exercise, the algebraic expression is \( a_n = \frac{n^2 - 1}{n^2 + 1} \). Here, \( n \) is a variable representing the position of a term in the sequence.
As \( n \) changes, the expression evaluates to different values.
- The numerator \( n^2 - 1 \) indicates that the sequence involves squared terms minus one.
- The denominator \( n^2 + 1 \) is similar but adds one instead.
Simplifying Fractions in Sequences
Fractions are an important part of understanding sequences, especially when dealing with expressions like \( \frac{n^2 - 1}{n^2 + 1} \). A fraction consists of a numerator and a denominator.
It represents a division of the numerator by the denominator.
When we calculate terms in a sequence using fractions, simplifying them makes results easier to understand.
It represents a division of the numerator by the denominator.
When we calculate terms in a sequence using fractions, simplifying them makes results easier to understand.
- Fraction simplification involves dividing both the numerator and the denominator by their greatest common divisor (GCD).
- For example, for \( a_3 = \frac{8}{10} \), the GCD is 2.
- Dividing both values by 2 gives \( \frac{4}{5} \).
Calculating Terms in a Sequence
Term calculation is the step-by-step process of finding specific terms from a sequence.
In sequences defined by algebraic expressions, calculating each term involves substituting a particular value of \( n \).
This method 'decodes' the general expression to find concrete numbers in the sequence pattern.
In sequences defined by algebraic expressions, calculating each term involves substituting a particular value of \( n \).
This method 'decodes' the general expression to find concrete numbers in the sequence pattern.
- For the first term \( (a_1) \), set \( n = 1 \) and solve: \( a_1 = \frac{0}{2} = 0 \).
- For the second term \( (a_2) \), with \( n = 2 \): \( a_2 = \frac{3}{5} \).
- Continuing this process generates each subsequent term by sequential substitution.
Using Mathematical Substitution
Mathematical substitution is a fundamental technique frequently used when working with algebraic expressions and sequences.
It involves replacing a variable in an expression with a specific value to compute a result.
In the context of sequences, substitution helps us find distinct terms.
It involves replacing a variable in an expression with a specific value to compute a result.
In the context of sequences, substitution helps us find distinct terms.
- In the sequence\( a_n = \frac{n^2 - 1}{n^2 + 1} \), \( n \) represents the term's position.
- Substituting values like \( n = 1, 2, 3, \) etc., calculates terms \( a_1, a_2, a_3 \), and so forth.
- Each accurate substitution transforms the algebraic expression into a hypothesis about a specific term of the sequence.
Other exercises in this chapter
Problem 10
Find \(a_{5}\) and \(a_{n}\) for each geometric sequence. $$a_{4}=18, r=2$$
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Use mathematical induction to prove each statement. Assume that \(n\) is a positive integer. $$1^{2}+2^{2}+3^{2}+\cdots+n^{2}=\frac{n(n+1)(2 n+1)}{6}$$
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Evaluate the following. In Exercises 17 and \(18,\) express the answer in terms of \(n .\) Do not use a calculator. $$\left(\begin{array}{c}12 \\\12\end{array}\
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Evaluate each expression. $$4 ! \cdot 5$$
View solution