Find the sum : 1.ⁿC₀ + 2.ⁿC₂ + 3.ⁿC₄ + 4.ⁿC₆ + ...........

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Please find below the solution to the asked query:

$$\begin{gathered} \mathrm{We} \mathrm{have}: \\ \mathrm{S}=1{.}^{\mathrm{n}}{\mathrm{C}}_0+2{.}^{\mathrm{n}}{\mathrm{C}}_2+3{.}^{\mathrm{n}}{\mathrm{C}}_4+4{.}^{\mathrm{n}}{\mathrm{C}}_6...... \\ \mathrm{It} \mathrm{can} \mathrm{be} \mathrm{written} \mathrm{as} \\ \mathrm{S}=\sum _{\mathrm{r}=0}^{\frac{\mathrm{n}}{2}}\left(\mathrm{r}+1\right).{ }^{\mathrm{n}}{\mathrm{C}}_{2\mathrm{r}} \\ =\sum _{\mathrm{r}=0}^{\frac{\mathrm{n}}{2}}{ }^{\mathrm{n}}{\mathrm{C}}_{2\mathrm{r}}+\sum _{\mathrm{r}=0}^{\frac{\mathrm{n}}{2}}\mathrm{r}.{ }^{\mathrm{n}}{\mathrm{C}}_{2\mathrm{r}} \\ =\left({}^{\mathrm{n}}{\mathrm{C}}_0{+}^{\mathrm{n}}{\mathrm{C}}_2{+}^{\mathrm{n}}{\mathrm{C}}_6+...\right)+\frac{1}{2}\sum _{\mathrm{r}=0}^{\frac{\mathrm{n}}{2}} \left(2\mathrm{r}\right).{ }^{\mathrm{n}}{\mathrm{C}}_{2\mathrm{r}} \\ =\left(\mathrm{Sum} \mathrm{of} \mathrm{even} \mathrm{binomail} \mathrm{coeficients}\right)+\frac{1}{2}\sum _{\mathrm{r}=0}^{\frac{\mathrm{n}}{2}} \left(\mathrm{n}{.}^{\mathrm{n}-1}{\mathrm{C}}_{2\mathrm{r}-1}\right) \\ \left[\mathrm{As} \mathrm{R}{.}^{\mathrm{N}}{\mathrm{C}}_{\mathrm{R}}=\mathrm{N}{.}^{\mathrm{N}-1}{\mathrm{C}}_{\mathrm{R}-1}\right] \\ =\frac{2^{\mathrm{n}}}{2}+\frac{\mathrm{n}}{2}\left[{}^{\mathrm{n}-1}{\mathrm{C}}_1{+}^{\mathrm{n}-1}{\mathrm{C}}_3{+}^{\mathrm{n}-1}{\mathrm{C}}_5+....\right] \\ =2^{\mathrm{n}-1}+\frac{\mathrm{n}}{2}\left[\mathrm{Sum} \mathrm{of} \mathrm{even} \mathrm{binomial} \mathrm{coefficients}\right] \\ =2^{\mathrm{n}-1}+\frac{\mathrm{n}}{2}\left[\frac{2^{\mathrm{n}-1}}{2}\right] \\ =2^{\mathrm{n}-1}+\mathrm{n}.2^{\mathrm{n}-3} \\ \Rightarrow 1{.}^{\mathrm{n}}{\mathrm{C}}_0+2{.}^{\mathrm{n}}{\mathrm{C}}_2+3{.}^{\mathrm{n}}{\mathrm{C}}_4+4{.}^{\mathrm{n}}{\mathrm{C}}_6......=2^{\mathrm{n}-1}+\mathrm{n}.2^{\mathrm{n}-3} \end{gathered}$$

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