Explanation Given info. 2 n > n Proof: Left hand side ( L . H . S . ) value of function P ( 1 ) = 2 1 = 2 R... To determine (b) To show: 2 n + 2 n > 2 n If P ( n ) is true for n = k then P ( n ) is true for n = k + 1 . Conclude that P ( n ) is true for all value of n.

4th Edition

ISBN: 9781319050733

Author: Rogawski

Publisher: MAC HIGHER

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Statements and Logic

Logic is the analysis of general patterns of thoughts without regard to any specific sense of context.

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Question

Chapter C, Problem 5E

To determine

(a)

To show:

$P (1)$ is true

To determine

(b)

To show:

$2^{n} + 2^{n} > 2 n$

If $P (n)$ is true for $n = k$ then $P (n)$ is true for $n = k + 1$ . Conclude that $P (n)$ is true for all value of n.

Expert Solution & Answer

Chapter C Solutions

CALCULUS (CLOTH)

Ch. C - Prob. 1E Ch. C - Prob. 2E Ch. C - Prob. 3E Ch. C - Prob. 4E Ch. C - Prob. 5E Ch. C - Prob. 6E Ch. C - Prob. 7E Ch. C - Prob. 8E Ch. C - Prob. 9E Ch. C - Prob. 10E

Ch. C - Prob. 11E Ch. C - Prob. 12E Ch. C - Prob. 13E Ch. C - Prob. 14E Ch. C - Prob. 15E Ch. C - Prob. 16E Ch. C - Prob. 17E Ch. C - Prob. 18E

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Show that if the statement is assumed to be true for , then it can be proved to be true for . Is the statement true for all positive integers ? Why?
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 can be proved to be true for n=k+1. Explain why this does not prove that the statement is true for all positive integers. Is the statement true for all positive integers? Why?

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(a) To show: P ( 1 ) is true

Solution Summary: The author explains that P ( 1 ) is true for the given condition.

Concept explainers

Videos

(a)

To show:

$P (1)$ is true

(b)

To show:

$2^{n} + 2^{n} > 2 n$

If $P (n)$ is true for $n = k$ then $P (n)$ is true for $n = k + 1$ . Conclude that $P (n)$ is true for all value of n.

Chapter C Solutions

(a) To show: P ( 1 ) is true

Solution Summary: The author explains that P ( 1 ) is true for the given condition.

Concept explainers

Videos

(a) To show: P(1) is true

(b) To show: 2n+2n>2n If P(n) is true for n=k then P(n) is true for n=k+1. Conclude that P(n) is true for all value of n.

Chapter C Solutions

(a)

To show:

$P (1)$ is true

(b)

To show:

$2^{n} + 2^{n} > 2 n$

If $P (n)$ is true for $n = k$ then $P (n)$ is true for $n = k + 1$ . Conclude that $P (n)$ is true for all value of n.