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: 37


The property was verified for $n=1$. The property was verified when $n$ was changed by $n+1$.

Work Step by Step

Prove the property for $n=1$ $(x_1)^{-1}=x_1^{-1}$ The property is verified for $n=1$ Assuming that $(x_1x_2x_3...x_n)^{-1}=x_1^{-1}x_2^{-1}x_3^{-1}...x_n^{-1}$ for all integers $n\geq1$, we need to prove that $(x_1x_2x_3...x_nx_{n+1})^{-1}=x_1^{-1}x_2^{-1}x_3^{-1}...x_n^{-1}x_{n+1}^{-1}$: $(x_1x_2x_3...x_nx_{n+1})^{-1}=(x_1x_2x_3...x_n)^{-1}x_{n+1}^{-1}=x_1^{-1}x_2^{-1}x_3^{-1}...x_n^{-1}x_{n+1}^{-1}$
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