 Prove the statements: If [n] is any odd integer, then (-1)^{n}=1.The proof of the given statement. allhvasstH 2020-11-08 Answered

Prove the statements: If n is any odd integer, then $$\displaystyle{\left(-{1}\right)}^{{{n}}}={1}$$.
The proof of the given statement.

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Consider the statement, If n is any odd integer, then $$\displaystyle{\left(-{1}\right)}^{{{n}}}={1}$$
The proof is given as,
Suppose n is any odd integer. By definition of odd integer, $$\displaystyle{n}={2}{p}\ +\ {1}$$ for some integer p.
$$\displaystyle{\left(-{1}\right)}^{{{n}}}={\left(-{1}\right)}^{{{2}{p}\ +\ {1}}}$$
$$\displaystyle={\left(-{1}\right)}^{{{2}{p}}}\ {\left(-{1}\right)}$$
$$\displaystyle={\left({1}\right)}^{{{p}}}\ {\left(-{1}\right)}$$
$$\displaystyle={1}\ \times\ {\left(-{1}\right)}$$
$$\displaystyle=\ -{1}$$
Therefore, if n an odd integer, then $$\displaystyle{\left(-{1}\right)}^{{{n}}}=\ -{1}$$.
Conclusion:
The statement, If n is any odd integer, the $$\displaystyle{\left(-{1}\right)}^{{{n}}}=\ -{1}$$ is proved.