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Exercise 2.9. Explain why the division theorem for a = 3 can be restated as follows: For a positive integer b, there exists a nonnegative integer q such that b equals either 3q, 3q + 1, or 3q + 2. Prove this result by induction, following the outline below: 1. First show that 1 can be written in the desired form. 2. Now suppose an integer b that is greater than or equal to 1 can be written in the desired form. Show that b+1 can also be so written. (Hint: There would appear to be three cases here, depending on whether b has the form 3q, 3q+1, or 3q+2, but really the first two can be combined into a single case. First assume thatb has the form 3q or 3g +1 for some nonnegative integer q and show that b+1 can be written in one of the three forms. Then assume that b has the form 3q + 2 for some nonnegative integer q and show that b+1 can be written in one of the three forms.) 3. Conclude by the principle of induction that every positive integer b can be written as 3q, 3q+ 1, or 3q + 2 for some nonnegative integer q. These two examples serve as models for a proof of the division theorem in general.

Exercise 2.9. Explain why the division theorem for a = 3 can be restated as follows: For a positive integer b, there exists a nonnegative integer q such that b equals either 3q, 3q + 1, or 3q + 2. Prove this result by induction, following the outline below: 1. First show that 1 can be written in the desired form. 2. Now suppose an integer b that is greater than or equal to 1 can be written in the desired form. Show that b+1 can also be so written. (Hint: There would appear to be three cases here, depending on whether b has the form 3q, 3q+1, or 3q+2, but really the first two can be combined into a single case. First assume thatb has the form 3q or 3g +1 for some nonnegative integer q and show that b+1 can be written in one of the three forms. Then assume that b has the form 3q + 2 for some nonnegative integer q and show that b+1 can be written in one of the three forms.) 3. Conclude by the principle of induction that every positive integer b can be written as 3q, 3q+ 1, or 3q + 2 for some nonnegative integer q. These two examples serve as models for a proof of the division theorem in general.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Having trouble with how this book wants me to do induction. There are more induction problems but a solid example of how to do it following this guide would help so much
20 2 Induction and the Division Theorem Exercise 2.9. Explain why the division theorem for a = 3 can be restated as follows: For a positive integer b, there exists a nonnegative integer q such that b equals either 3q, 3q+ 1, or 3q+2. Prove this result by induction, following the outline below: 1. First show that 1 can be written in the desired form. 2. Now suppose an integer b that is greater than or equal to 1 can be written in the desired form. Show that b+1 can also be so written. (Hint: There would appear to be three cases here, depending on whether b has the form 3q, 3q+1, or 3q+2, but really the first two can be combined into a single case. First assume that b has the form 3q or 3q + 1 for some nonnegative integer q and show that b+1 can be written in one of the three forms. Then assume that b has the form 3g +2 for some nonnegative integer q and show that b+1 can be written in one of the three forms.) 3. Conclude by the principle of induction that every positive integer b can be written as 3q, 3q+ 1, or 3q+2 for some nonnegative integer q. These two examples serve as models for a proof of the division theorem in general.
Transcribed Image Text:20 2 Induction and the Division Theorem Exercise 2.9. Explain why the division theorem for a = 3 can be restated as follows: For a positive integer b, there exists a nonnegative integer q such that b equals either 3q, 3q+ 1, or 3q+2. Prove this result by induction, following the outline below: 1. First show that 1 can be written in the desired form. 2. Now suppose an integer b that is greater than or equal to 1 can be written in the desired form. Show that b+1 can also be so written. (Hint: There would appear to be three cases here, depending on whether b has the form 3q, 3q+1, or 3q+2, but really the first two can be combined into a single case. First assume that b has the form 3q or 3q + 1 for some nonnegative integer q and show that b+1 can be written in one of the three forms. Then assume that b has the form 3g +2 for some nonnegative integer q and show that b+1 can be written in one of the three forms.) 3. Conclude by the principle of induction that every positive integer b can be written as 3q, 3q+ 1, or 3q+2 for some nonnegative integer q. These two examples serve as models for a proof of the division theorem in general.
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