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For \( n \geq 2 \), show that
\[
\begin{array}{l}
{\left[1+\left(\frac{1+i}{2}\right)\right]\left[1+\left(\frac{1+i}{2}\right)^{2}\right]\left[1+\left(\frac{1+i}{2}\right)^{2^{2}}\right]} \\
\quad \cdots\left[1+\left(\frac{1+i}{2}\right)^{2^{n}}\right]=(1+i)\left(1-\frac{1}{2^{2^{n}}}\right) \text {, where } i=\sqrt{-1} .
\end{array}
\]
\( \mathrm{P} \)
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