Algebra and Trigonometry 10th Edition

Published by Cengage Learning
ISBN 10: 9781337271172
ISBN 13: 978-1-33727-117-2

Chapter 11 - 11.4 - Mathematical Induction - 11.4 Exercises - Page 806: 33


The property was proved for $n=1$ The property is correct if n is changed by $n+1$

Work Step by Step

Let's prove the property for $n=1$: $2^{2n+1}+1=2^{2(1)+1}+1=2^5+1=32+1=33=3(11)$ 3 is a factor. It is correct! Suppose that the property is correct, that is: $2^{2n+1}+1$ has 3 as a factor for all integers $n\geq1$ Now, let's prove the property for $n+1$: $2^{2(n+1)+1}+1=2^{2n+3}+1=2^{2+2n+1}+1=2^2·2^{2n+1}+1=4·2^{2n+1}+1=(3+1)2^{2n+1}+1=3·2^{2n+1}+(2^{2n+1}+1)$ It is a sum of two terms that have 3 as a factor.
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