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If \( (1+x)^{n}=C_{0}+C_{1} x+C_{2} x^{2}+C_{3} x^{3}+C_{n} x^{n} \), prove
\( \mathrm{P} \)
that
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Prove that \( C_{0}+\left(C_{0}+C_{1}\right)+\left(C_{0}+C_{1}+C_{2}\right)+\ldots\left(C_{0}+\right. \) \( \left.C_{1}+C_{2}+\ldots+C_{n-1}\right)=n 2^{n-1} \) (where \( n \) is an even integer).
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