If \( C_{0}, C_{1}, C_{2}, \ldots C_{n} \) are the binomial coeffic...

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If \( C_{0}, C_{1}, C_{2}, \ldots C_{n} \) are the binomial coefficients in the expansion of \( (1+x)^{n} \) then prove that
\( \mathrm{P} \)
W
\[
\frac{2^{2} \cdot C_{0}}{1 \cdot 2}+\frac{2^{3} \cdot C_{1}}{2 \cdot 3}+\frac{2^{4} \cdot C_{2}}{3 \cdot 4}+\ldots \ldots+\frac{2^{n+2} \cdot C_{n}}{(n+1)(n+2)}=\frac{3^{n+2}-2 n-5}{(n+1)(n+2)}
\]
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