It is a fact that if n is any nonnegative integer, then 1-(1/2"+1) 1 1 1 1+=+ + + •+ 2 2² 23 1-(1/2) 1 2" (A more general form of this statement is proved in Section 5.2.) Is the right-hand side of this equation rational? If so, express it as a ratio of two integers.
It is a fact that if n is any nonnegative integer, then 1-(1/2"+1) 1 1 1 1+=+ + + •+ 2 2² 23 1-(1/2) 1 2" (A more general form of this statement is proved in Section 5.2.) Is the right-hand side of this equation rational? If so, express it as a ratio of two integers.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
27
![### Educational Assessment Questions
**21.** True or false? If \( m \) is any even integer and \( n \) is any odd integer, then \( m^2 + 3n \) is odd. Explain.
**22.** True or false? If \( a \) is any odd integer, then \( a^2 + a \) is even. Explain.
**23.** True or false? If \( k \) is any even integer and \( m \) is any odd integer, then \( (k+2)^2 - (m-1)^2 \) is even. Explain.
### Derived Statements
Derive the statements in questions 24–26 as corollaries of Theorems 4.3.1, 4.3.2, and the results of exercises 12, 13, 14, 15, and 17.
**24.** For any rational numbers \( r \) and \( s \), \( 2r + 3s \) is rational.
**25.** If \( r \) is any rational number, then \( 3r^2 - 2r + 4 \) is rational.
**26.** For any rational number \( s \), \( 5s^3 + 8s^2 - 7 \) is rational.
**27.** It is a fact that if \( n \) is any nonnegative integer, then
\[
1 + \frac{1}{2} + \frac{1}{2^2} + \frac{1}{2^3} + \cdots + \frac{1}{2^n} = \frac{1 - (1/2^{n+1})}{1 - (1/2)}
\]
(A more general form of this statement is proved in Section 5.2.) Is the right-hand side of this equation rational? If so, express it as a ratio of two integers.
**28.** Suppose \( a, b, c, \) and \( d \) are integers and \( a \neq c \). Suppose also that \( x \) is a real number that satisfies the equation
\[
\frac{ax + b}{cx + d} = 1.
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8ec5dee8-4b81-4c1e-b6ca-023c2626018b%2Fbbe070ec-1647-4400-9fbb-a0ed2803f61f%2Fthznt6_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Educational Assessment Questions
**21.** True or false? If \( m \) is any even integer and \( n \) is any odd integer, then \( m^2 + 3n \) is odd. Explain.
**22.** True or false? If \( a \) is any odd integer, then \( a^2 + a \) is even. Explain.
**23.** True or false? If \( k \) is any even integer and \( m \) is any odd integer, then \( (k+2)^2 - (m-1)^2 \) is even. Explain.
### Derived Statements
Derive the statements in questions 24–26 as corollaries of Theorems 4.3.1, 4.3.2, and the results of exercises 12, 13, 14, 15, and 17.
**24.** For any rational numbers \( r \) and \( s \), \( 2r + 3s \) is rational.
**25.** If \( r \) is any rational number, then \( 3r^2 - 2r + 4 \) is rational.
**26.** For any rational number \( s \), \( 5s^3 + 8s^2 - 7 \) is rational.
**27.** It is a fact that if \( n \) is any nonnegative integer, then
\[
1 + \frac{1}{2} + \frac{1}{2^2} + \frac{1}{2^3} + \cdots + \frac{1}{2^n} = \frac{1 - (1/2^{n+1})}{1 - (1/2)}
\]
(A more general form of this statement is proved in Section 5.2.) Is the right-hand side of this equation rational? If so, express it as a ratio of two integers.
**28.** Suppose \( a, b, c, \) and \( d \) are integers and \( a \neq c \). Suppose also that \( x \) is a real number that satisfies the equation
\[
\frac{ax + b}{cx + d} = 1.
\]
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 2 steps with 2 images
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
