Discrete Mathematics with Applications 4th Edition

Published by Cengage Learning
ISBN 10: 0-49539-132-8
ISBN 13: 978-0-49539-132-6

Chapter 4 - Elementary Number Theory and Methods of Proof - Exercise Set 4.1 - Page 162: 34

Answer

$\begin{split} (-1)^n & = (-1)(-1)(-1)(-1)...(-1) \\ & = (-1)^2(-1)^2...(-1) \\ & = 1(-1) \\ & = -1 \\ \end{split}$

Work Step by Step

Let $n$ be an odd integer. $a$ can't be equal to zero, because zero is even $(0 = 2.0)$ If a \geqslant 0: $(-1)^n = (-1)^1(-1)^1(-1)^1(-1)^1...(-1)$ $n$ times, but $n$ is odd, so we can group the pairs: $(-1)^n = (-1)^2(-1)^2...(-1)$ and we'll have 1 left. Knowing that $(-1)^2 = 1$, we can do: $\begin{split} (-1)^n & = 1.1.1.1...(-1) \\ & = -1 \\ \end{split}$ If a < 0 (similar): $(-1)^n = (-1)^{-1}(-1)^{-1}(-1)^{-1}(-1)^{-1}...(-1)^{-1}$ But $(-1)^{-1} = \dfrac{-1}{1} = -1 = (-1)^1$, now repeat the same process we did when a > 0. So $(-1)^n$ (with $n$ being and odd number), is always equals to -1.
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