Prove that $(1 + i)^{2 n} + (1 - i)^{2 n}$ is a real number, for $n \in N$ .

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Solution

Now,
$(1 + i)^{2 n} + (1 - i)^{2 n}$
$= (1 + i)^{2 n} + ¯ ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ ¯ (1 + i)^{2 n}$ [ Since $(1 - i) = ¯ ¯¯¯¯¯¯¯¯¯¯¯¯¯¯ ¯ (1 + i)$ ]
$= 2 R e {(1 + i)^{2 n}}$ [ Since $z + ¯ ¯ ¯ z = 2 R e (z)$ ], which is a real number.

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