Put the following statements into order to prove that if 3n+4 is even then n is even. Put N next to the 3 statements that should not be used. 1. Then, 3n+4=3(2k+1)+4=6k+7=2(3k+3)+1. 2. By definition of odd, there exists integer k such that n=2k+1. 3. Thus, if n is odd then 3n+4 is odd or by contradiction, if 3n+4 is even then n is even,. 4. Therefore, by definition of odd, 3n+4 is odd. 5. Since k is an integer, t =3k+3 is also an integer. 6. Thus, there exists an integer t such that 3n+4=2t+1. 7. Suppose n is odd. 8. Thus, if n is odd then 3n+4 is odd or by contraposition, if 3n+4 is even then n is even, . 9. Suppose 3n+4 is even. 10. By definition of odd, n=2k+1.
Put the following statements into order to prove that if 3n+4 is even then n is even. Put N next to the 3 statements that should not be used. 1. Then, 3n+4=3(2k+1)+4=6k+7=2(3k+3)+1. 2. By definition of odd, there exists integer k such that n=2k+1. 3. Thus, if n is odd then 3n+4 is odd or by contradiction, if 3n+4 is even then n is even,. 4. Therefore, by definition of odd, 3n+4 is odd. 5. Since k is an integer, t =3k+3 is also an integer. 6. Thus, there exists an integer t such that 3n+4=2t+1. 7. Suppose n is odd. 8. Thus, if n is odd then 3n+4 is odd or by contraposition, if 3n+4 is even then n is even, . 9. Suppose 3n+4 is even. 10. By definition of odd, n=2k+1.
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter2: The Integers
Section2.2: Mathematical Induction
Problem 50E: Show that if the statement 1+2+3+...+n=n(n+1)2+2 is assumed to be true for n=k, the same equation...
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Transcribed Image Text:Put the following statements into order to prove that if 3n+4 is even then n is even. Put N next to the 3 statements that
should not be used.
1. Then, 3n+4=3(2k+1)+4=6k+7=2(3k+3)+1.
2. By definition of odd, there exists integer k such that n=2k+1.
3. Thus, if n is odd then 3n+4 is odd or by contradiction, if 3n+4 is even then n is even,.
4. Therefore, by definition of odd, 3n+4 is odd.
5. Since k is an integer, t =3k+3 is also an integer.
6. Thus, there exists an integer t such that 3n+4=2t+1.
7. Suppose n is odd.
8. Thus, if n is odd then 3n+4 is odd or by contraposition, if 3n+4 is even then n is even, .
9. Suppose 3n+4 is even.
10. By definition of odd, n=2k+1.
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