Explanation Given information: n is a positive integer. Proof: The hypothesis is actually strong inductive hypothesis, but we really don't need it. Only by induction also it can be proved. 4 1 = 4, 4 2 = 16, 4 3 = 64, 4 4 = 256, 4 5 = 1024, 4 6 = 4096, 4 7 = 16384, 4 8 = 65536. Conjecture: The units digit of 4 n is 4 if n is an odd integer and it is 6 if n is an even integer, for all integers n > 1. Let us denote the conjecture by P ( n ). for the base case n = 1, it is trivial that the units place of 4 1 is 4 and n = 1 is odd. So, P (1) is true. For any integer k > 1, assume that for all integers l, 1 ≤ l ≤ k , P ( l ) is true. We need to prove that P ( k + 1 ) is also true. Case 1: k is odd. 4 k +1 = 4 k .4. by induction hypothesis, the units place of 4 k is 4, so, 4 k = 4 + 10m, where m is some integer ( for example 4 3 = 4 + 10 × 6 )

5th Edition

ISBN: 9780357540244

Author: EPP

Publisher: CENGAGE L

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Chapter 5.4, Problem 17ES

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Compute 4¹, 4², 4³, 4⁴, 4⁵, 4⁶, 4⁷, and 4⁸. Make a conjecture about the units digit of 4n where n is a positive integer. Use strong mathematical induction to prove your conjecture.

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Chapter 5.4, Problem 16ES

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To prove: Compute 4 1 , 4 2 , 4 3 , 4 4 , 4 5 , 4 6 , 4 7 , and 4 8 . Make a conjecture about the units digit of 4n where n is a positive integer. Use strong mathematical induction to prove your conjecture.

Solution Summary: The author explains how to prove a conjecture about the units digit of 4n where n is an odd integer.

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To prove:

Compute 4¹, 4², 4³, 4⁴, 4⁵, 4⁶, 4⁷, and 4⁸. Make a conjecture about the units digit of 4n where n is a positive integer. Use strong mathematical induction to prove your conjecture.

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Chapter 5 Solutions

To prove: Compute 4 1 , 4 2 , 4 3 , 4 4 , 4 5 , 4 6 , 4 7 , and 4 8 . Make a conjecture about the units digit of 4n where n is a positive integer. Use strong mathematical induction to prove your conjecture.

Solution Summary: The author explains how to prove a conjecture about the units digit of 4n where n is an odd integer.

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To prove: Compute 41, 42, 43, 44, 45, 46, 47, and 48. Make a conjecture about the units digit of 4n where n is a positive integer. Use strong mathematical induction to prove your conjecture.

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Chapter 5 Solutions

To prove:

Compute 4¹, 4², 4³, 4⁴, 4⁵, 4⁶, 4⁷, and 4⁸. Make a conjecture about the units digit of 4n where n is a positive integer. Use strong mathematical induction to prove your conjecture.