Let S = frac{2}{1} {}^{n}C_{0} + frac{2^{2}}{ | vedclass

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(B) We know that $(1+x)^{n} = {}^{n}C_{0} + x{}^{n}C_{1} + x^{2}{}^{n}C_{2} + \ldots + x^{n}{}^{n}C_{n}$.Integrating both sides with respect to $x$ fr

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$\frac{3^{n+1}-1}{n+1}$

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(B) We know that $(1+x)^{n} = {}^{n}C_{0} + x{}^{n}C_{1} + x^{2}{}^{n}C_{2} + \ldots + x^{n}{}^{n}C_{n}$.Integrating both sides with respect to $x$ from $0$ to $2$, we get:$\int_{0}^{2} (1+x)^{n} dx = \int_{0}^{2} \left( {}^{n}C_{0} + x{}^{n}C_{1} + x^{2}{}^{n}C_{2} + \ldots + x^{n}{}^{n}C_{n} \right) dx$.Evaluating the integral on the left side:$\left[ \frac{(1+x)^{n+1}}{n+1} \right]_{0}^{2} = \frac{(1+2)^{n+1}}{n+1} - \frac{(1+0)^{n+1}}{n+1} = \frac{3^{n+1}-1}{n+1}$.Evaluating the integral on the right side:$\left[ x{}^{n}C_{0} + \frac{x^{2}}{2}{}^{n}C_{1} + \frac{x^{3}}{3}{}^{n}C_{2} + \ldots + \frac{x^{n+1}}{n+1}{}^{n}C_{n} \right]_{0}^{2} = \frac{2}{1}{}^{n}C_{0} + \frac{2^{2}}{2}{}^{n}C_{1} + \frac{2^{3}}{3}{}^{n}C_{2} + \ldots + \frac{2^{n+1}}{n+1}{}^{n}C_{n}$.Thus, $S = \frac{3^{n+1}-1}{n+1}$.

Let $S = \frac{2}{1} {}^{n}C_{0} + \frac{2^{2}}{2} {}^{n}C_{1} + \frac{2^{3}}{3} {}^{n}C_{2} + \ldots + \frac{2^{n+1}}{n+1} {}^{n}C_{n}$. Then, $S$ equals

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