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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ensojadasH
Answered 2020-11-09 Author has 15671 answers
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.
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