Statement. For all n = Z2o, n² - 5n is even. Proof. Let P(n) be the statement that n² - 5n is even. We will prove that P(n) is true for all n € Zzo. For our base case where n = = 0, we have (0)² – 5(0) = 0, which is even. So P(0) is true. For our induction step, suppose P(k) is true for some k ≥ 0; that is, k² – 5k is even. We will show that P(k+1) is true; that is, (k+ 1)² − 5(k + 1) is even. Observe: (k+ 1)² - 5(k+ 1) = k² + 2k +1 − 5k - 5 = k² - 3k - 4 = = (k+ 1)(k-4) So we can show P(k+ 1) is true by showing (k+1)(k − 4) is even. Since k is a nonnegative integer, k is also an integer. If k is even, then by definition, k = 2r for some re Z. Then (k+ 1)(k − 4) = (2r + 1)(2r — 4) = 2(2r + 1)(r − 2). Since r is an integer, (2r + 1)(r − 2) is an integer, and by definition, (k+ 1)(k − 4) is even. If k is odd, then by definition, k = 2q + 1 for some q € Z. Then (k+ 1)(k − 4) = (2q + 2)(2q − 3) = 2(q + 1) (2q 3). Since q is an integer, (q + 1)(2q − 3) is an integer, and by definition (k+ 1)(k − 4) is even. So P(k+ 1) is true. Therefore, by the Principle of Mathematical Induction, P(n) is true for all n € Zzo (a) Do you believe the statement is true? If so why? If not, give an examlpe to show why not. (b) There is something wrong with the proof. EXPLAIN what it is. (c) What kind of induction is being used? Strong or weak? How do you know?
Statement. For all n = Z2o, n² - 5n is even. Proof. Let P(n) be the statement that n² - 5n is even. We will prove that P(n) is true for all n € Zzo. For our base case where n = = 0, we have (0)² – 5(0) = 0, which is even. So P(0) is true. For our induction step, suppose P(k) is true for some k ≥ 0; that is, k² – 5k is even. We will show that P(k+1) is true; that is, (k+ 1)² − 5(k + 1) is even. Observe: (k+ 1)² - 5(k+ 1) = k² + 2k +1 − 5k - 5 = k² - 3k - 4 = = (k+ 1)(k-4) So we can show P(k+ 1) is true by showing (k+1)(k − 4) is even. Since k is a nonnegative integer, k is also an integer. If k is even, then by definition, k = 2r for some re Z. Then (k+ 1)(k − 4) = (2r + 1)(2r — 4) = 2(2r + 1)(r − 2). Since r is an integer, (2r + 1)(r − 2) is an integer, and by definition, (k+ 1)(k − 4) is even. If k is odd, then by definition, k = 2q + 1 for some q € Z. Then (k+ 1)(k − 4) = (2q + 2)(2q − 3) = 2(q + 1) (2q 3). Since q is an integer, (q + 1)(2q − 3) is an integer, and by definition (k+ 1)(k − 4) is even. So P(k+ 1) is true. Therefore, by the Principle of Mathematical Induction, P(n) is true for all n € Zzo (a) Do you believe the statement is true? If so why? If not, give an examlpe to show why not. (b) There is something wrong with the proof. EXPLAIN what it is. (c) What kind of induction is being used? Strong or weak? How do you know?
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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Transcribed Image Text:Statement. For all n € Z≥o, n² – 5n is even.
-
Proof. Let P(n) be the statement that n². 5n is even. We will prove that P(n) is true for all n € Z≥o.
For our base case where n = 0, we have (0)² - 5(0) = 0, which is even. So P(0) is true.
For our induction step, suppose P(k) is true for some k ≥ 0; that is, k² – 5k is even. We will show
that P(k+ 1) is true; that is, (k+ 1)² − 5(k + 1) is even. Observe:
(k+ 1)² - 5(k+ 1) = k² + 2k +1 − 5k – 5
= k² - 3k - 4
= (k + 1)(k − 4)
So we can show P(k+ 1) is true by showing (k+1)(k − 4) is even.
Since k is a nonnegative integer, k is also an integer. If k is even, then by definition, k = 2r for some
r € Z. Then (k+ 1)(k − 4) = (2r + 1)(2r − 4) = 2(2r + 1) (r − 2). Since r is an integer, (2r + 1)(r − 2)
is an integer, and by definition, (k+ 1)(k − 4) is even.
If k is odd, then by definition, k = 2q + 1 for some q € Z. Then (k + 1)(k − 4) = (2q + 2)(2q − 3) :
2(q + 1) (2q - 3). Since q is an integer, (q + 1)(2q − 3) is an integer, and by definition (k+ 1)(k − 4) is
even.
So P(k+1) is true. Therefore, by the Principle of Mathematical Induction, P(n) is true for all
n € Zzo
E
1
(a) Do you believe the statement is true? If so why? If not, give an examlpe to show why not.
(b) There is something wrong with the proof. EXPLAIN what it is.
(c) What kind of induction is being used? Strong or weak? How do you know?
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