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