Problem 61
Question
Write an expression for the \(n\) th term of the sequence. (There is more than one correct answer.) \(1,-\frac{1}{1 \cdot 3}, \frac{1}{1 \cdot 3 \cdot 5},-\frac{1}{1 \cdot 3 \cdot 5 \cdot 7}, \ldots\)
Step-by-Step Solution
Verified Answer
The nth term of the sequence can be expressed as \(a_n = (-1)^{n+1} \cdot \frac{1}{(2n-1)!!}\)
1Step 1: Identify the sequence pattern
Look at the pattern given. Notice that the denominators are increasing products of odd numbers, and the signs of the terms alternate. All denominators have odd numbers and there is no common difference. This suggests that the expression involves factorials or a variation of factorial.
2Step 2: Express the pattern using mathematical notation
Given that if n = 1, we start with 1, and then start multiplying by successive odd numbers as n increases. We can write the denominator as the product of first n odd numbers. The alternating sign can be expressed as \(-1^{n+1}\). The term is then expressed as \(a_n= (-1)^{n+1} \cdot \frac{1}{(2n-1)!!}\) where \((2n-1)!!\) represents the double factorial of odd numbers, which means multiplying together the sequence of numbers with a common difference of 2. If n=1, there is no denominator which aligns with our sequence.
3Step 3: Verification
Verify the formula by substituting the first few values of n. If the terms match with the given sequence, then the formula for the nth term is correct.
Key Concepts
Alternating SequencesDouble FactorialsMathematical Patterns
Alternating Sequences
In mathematics, alternating sequences are series of numbers in which the sign changes between each term. This means the sequence alternates between positive and negative values. These sequences often appear in mathematical problems and can be identified by a specific pattern in their terms.
For example:
To mathematically express this alternation, we use the expression \((-1)^{n+1}\). Here, the exponent determines the sign of each term. When \(n+1\) is even, the result is positive; when it's odd, the result is negative.
Recognizing this pattern allows us to write sequences that switch signs predictably, making them valuable for solving series and other mathematical problems.
For example:
- The sequence given in the exercise is: \(1, -\frac{1}{1 \cdot 3}, \frac{1}{1 \cdot 3 \cdot 5}, -\frac{1}{1 \cdot 3 \cdot 5 \cdot 7}, \ldots\)
To mathematically express this alternation, we use the expression \((-1)^{n+1}\). Here, the exponent determines the sign of each term. When \(n+1\) is even, the result is positive; when it's odd, the result is negative.
Recognizing this pattern allows us to write sequences that switch signs predictably, making them valuable for solving series and other mathematical problems.
Double Factorials
The double factorial is a concept that extends the idea of a regular factorial. While a regular factorial \(n!\) is the product of all positive integers up to \(n\), the double factorial uses only every other integer. This can be applied either to even or odd numbers.
For odd numbers, the double factorial \((2n-1)!!\) is the product of all odd numbers from 1 up to \((2n-1)\).
The notation \((2n-1)!!\) explicitly refers to taking these odd-numbered products, which aligns perfectly with the pattern we identified in the exercise.
Understanding double factorials is crucial, especially when dealing with sequences that involve non-standard progressions or specific patterns.
For odd numbers, the double factorial \((2n-1)!!\) is the product of all odd numbers from 1 up to \((2n-1)\).
- For example, \((5)!! = 5 \cdot 3 \cdot 1 = 15\).
The notation \((2n-1)!!\) explicitly refers to taking these odd-numbered products, which aligns perfectly with the pattern we identified in the exercise.
Understanding double factorials is crucial, especially when dealing with sequences that involve non-standard progressions or specific patterns.
Mathematical Patterns
Mathematical patterns help identify and generalize the behavior of sequences. In the context of this exercise, identifying patterns was key to formulating the expression for the nth term.
Let's summarize these patterns:
Such patterns can apply to a wide variety of mathematical contexts, including series, functions, and even real-world phenomena where predictable behavior is observed. The ability to discern and articulate these patterns is a foundational skill in mathematics, enhancing both comprehension and problem-solving capabilities.
Let's summarize these patterns:
- **Alternating signs**: Each term alternates between positive and negative, which required the use of \((-1)^{n+1}\).
- **Increasing products of odd numbers**: The denominators increase through a product of consecutive odd numbers, captured by the double factorial notation.
Such patterns can apply to a wide variety of mathematical contexts, including series, functions, and even real-world phenomena where predictable behavior is observed. The ability to discern and articulate these patterns is a foundational skill in mathematics, enhancing both comprehension and problem-solving capabilities.
Other exercises in this chapter
Problem 60
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