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: 32


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 $n=1$: $n^4-n+4=1^4-1+4=4=2(2)$ 2 is a factor. It is correct! Suppose that the property is correct, that is: $n^4-n+4$ has 2 as a factor for all integers $n\geq1$ Now, let's prove the property for $n+1$: $(n+1)^4-(n+1)+4=n^4+4n^3+6n^2+4n+1-n-1+4=(n^4-n+4)+(4n^3+6n^2+4n+1-1)=(n^4-n+4)+2(2n^3+3n^2+2n)$ It is a sum of two terms that have 2 as a factor.
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