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: 34


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