Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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What is the 9th term of this sequence ?
could you show the process please?
![Certainly! The image contains a mathematical expression, which is transcribed below and explained for educational purposes:
### Mathematical Expression:
\[ a_n = (-1)^{2n} (n - 10)_2 \]
### Explanation:
1. **\( a_n \)**: This denotes the \( n \)-th term in a sequence.
2. **\( (-1)^{2n} \)**: This term involves \( -1 \) raised to the power of \( 2n \). Since \( 2n \) is always even, \( (-1)^{2n} \) simplifies to \( 1 \), as any even power of \(-1\) results in \( 1 \).
3. **\( (n - 10)_2 \)**: This appears to be a mathematical construct known as the Pochhammer symbol or shifted factorial, defined as \( (n - 10)(n - 11) \) in this context. This notation is often used in combinatorics and special functions.
### Simplification:
Given that \( (-1)^{2n} = 1 \), the expression simplifies to:
\[ a_n = 1 \cdot (n - 10)(n - 11) \]
Thus:
\[ a_n = (n - 10)(n - 11) \]
### Summary:
The given formula calculates the \( n \)-th term of a sequence by evaluating the product of \( n - 10 \) and \( n - 11 \). The notation \( (-1)^{2n} \) consistently equals 1 due to the exponent being an even number, hence doesn't affect the final product. The term \( (n - 10)_2 \) represents the product of two consecutive terms starting from \( n - 10 \) and \( n - 11 \).
This formula can be useful in various mathematical contexts including sequences, series, and combinatorial problems.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff1236809-23f1-4bf3-a287-e517d49ba79e%2F5bde041e-158f-4302-9e8f-8f5bcb0df9aa%2Fx1khnpq.jpeg&w=3840&q=75)
Transcribed Image Text:Certainly! The image contains a mathematical expression, which is transcribed below and explained for educational purposes:
### Mathematical Expression:
\[ a_n = (-1)^{2n} (n - 10)_2 \]
### Explanation:
1. **\( a_n \)**: This denotes the \( n \)-th term in a sequence.
2. **\( (-1)^{2n} \)**: This term involves \( -1 \) raised to the power of \( 2n \). Since \( 2n \) is always even, \( (-1)^{2n} \) simplifies to \( 1 \), as any even power of \(-1\) results in \( 1 \).
3. **\( (n - 10)_2 \)**: This appears to be a mathematical construct known as the Pochhammer symbol or shifted factorial, defined as \( (n - 10)(n - 11) \) in this context. This notation is often used in combinatorics and special functions.
### Simplification:
Given that \( (-1)^{2n} = 1 \), the expression simplifies to:
\[ a_n = 1 \cdot (n - 10)(n - 11) \]
Thus:
\[ a_n = (n - 10)(n - 11) \]
### Summary:
The given formula calculates the \( n \)-th term of a sequence by evaluating the product of \( n - 10 \) and \( n - 11 \). The notation \( (-1)^{2n} \) consistently equals 1 due to the exponent being an even number, hence doesn't affect the final product. The term \( (n - 10)_2 \) represents the product of two consecutive terms starting from \( n - 10 \) and \( n - 11 \).
This formula can be useful in various mathematical contexts including sequences, series, and combinatorial problems.
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