Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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For every integer n ≥ 2, let P(n) be the following inequality. 2n< (n + 1)! (a) What is P(2)? O 4 < (n + 2)! 4 <4 4 < 6 4 <2 O 2 < (n + 2)! Is P(2) true? Yes O No (b) What is P(k)? Ok² < (K + 1)! Ok² <k! O 2k < (k + 1)! O 2k <k! 2n< (n + k)! (c) What is P(k + 1)? O 2k + 1 < (k+ 1)! O 2 + 1 < (n + k + 1)! O 2k +1 < (k + 2)! O (k+ 1)² < (k+ 1)! ○ (k+ 1)² < (k+ 2)! (d) In a proof by mathematical induction that this inequality holds for every integer n ≥ 2, what must be shown in the inductive step? O We need to show that if k is any integer with k ≥ 2 and if P(k) is true, then P(k+ 1) is also true. O We need to show that if k is any integer with k ≥ 2 and if P(k) is true, then P(k + 1) is false. O We need to show that if k is any integer with k ≥ 2 and if P(k) is false, then P(k + 1) is true. O We need to show that if k is any integer with k ≥ 2 and if P(k) is false, then P(k+ 1) is also false.
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Transcribed Image Text:For every integer n ≥ 2, let P(n) be the following inequality. 2n< (n + 1)! (a) What is P(2)? O 4 < (n + 2)! 4 <4 4 < 6 4 <2 O 2 < (n + 2)! Is P(2) true? Yes O No (b) What is P(k)? Ok² < (K + 1)! Ok² <k! O 2k < (k + 1)! O 2k <k! 2n< (n + k)! (c) What is P(k + 1)? O 2k + 1 < (k+ 1)! O 2 + 1 < (n + k + 1)! O 2k +1 < (k + 2)! O (k+ 1)² < (k+ 1)! ○ (k+ 1)² < (k+ 2)! (d) In a proof by mathematical induction that this inequality holds for every integer n ≥ 2, what must be shown in the inductive step? O We need to show that if k is any integer with k ≥ 2 and if P(k) is true, then P(k+ 1) is also true. O We need to show that if k is any integer with k ≥ 2 and if P(k) is true, then P(k + 1) is false. O We need to show that if k is any integer with k ≥ 2 and if P(k) is false, then P(k + 1) is true. O We need to show that if k is any integer with k ≥ 2 and if P(k) is false, then P(k+ 1) is also false.
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