Discrete Mathematics with Applications 4th Edition

Published by Cengage Learning
ISBN 10: 0-49539-132-8
ISBN 13: 978-0-49539-132-6

Chapter 5 - Sequences, Mathematical Induction, and Recursion - Exercise Set 5.3 - Page 266: 17

Answer

proof by mathematical induction provided below

Work Step by Step

proof by mathematical induction: suppose P(n) = 1 + 3n ≤ 4^n, for every integer n ≥ 0. Basis step: Show that P(0) is true: P(0) = 1+ 3(0) = 1 ≤ 1 (True) Inductive Step: Show that for all integers K ≥ 0, if P(K) is true then P(K+1) is true: suppose P(K) = 1+3K ≤ 4^K is true (Inductive hypothesis) P(K+1) = 1 + 3(k+1) ≤ 4^(K+1) = 1 + 3k + 3 ≤ 4 (4^K) but 1 + 3k is ≤ 4^k ( by P(K) ), so it yields: (1 + 3K) + 3 ≤ 4^k + 3 (4^K) but 3 ≤ 3 (4^K) for all K ≥ 0 so (1 + 3K) + 3 ≤ 4^(K+1) is true therefore, P(n) is true for all n ≥ 0
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