Hurry tell this ques by writing all statements

Hurry tell this ques by writing all statements 0

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Please find below the solution to the asked query:

$$\begin{gathered} \mathrm{We} \mathrm{know} \mathrm{that} \\ {\left(1+\mathrm{x}\right)}^{\mathrm{n}}{=}^{\mathrm{n}}{\mathrm{C}}_0{+}^{\mathrm{n}}{\mathrm{C}}_1\mathrm{x}{+}^{\mathrm{n}}{\mathrm{C}}_2{\mathrm{x}}^2+...{+}^{\mathrm{n}}{\mathrm{C}}_{\mathrm{n}}{\mathrm{x}}^{\mathrm{n}} \\ \mathrm{Integrate} \mathrm{both} \mathrm{sides} \mathrm{from} 0 \mathrm{to} 3 \\ {\left[\frac{{\left(1+\mathrm{x}\right)}^{\mathrm{n}+1}}{\mathrm{n}+1}\right]}_0^3={\left[{}^{\mathrm{n}}{\mathrm{C}}_0\mathrm{x}+\frac{{}^{\mathrm{n}}{\mathrm{C}}_1{\mathrm{x}}^2}{2}+\frac{{}^{\mathrm{n}}{\mathrm{C}}_2{\mathrm{x}}^3}{3}+....+\frac{{}^{\mathrm{n}}{\mathrm{C}}_{\mathrm{n}}{\mathrm{x}}^{\mathrm{n}+1}}{\mathrm{n}+1}\right]}_0^3 \\ \Rightarrow \frac{4^{\mathrm{n}+1}}{\mathrm{n}+1}-\frac{1}{\mathrm{n}+1}{=}^{\mathrm{n}}{\mathrm{C}}_{0}3+\frac{{}^{\mathrm{n}}{\mathrm{C}}_1{3}^2}{2}+\frac{{}^{\mathrm{n}}{\mathrm{C}}_2{3}^3}{3}+....+\frac{{}^{\mathrm{n}}{\mathrm{C}}_{\mathrm{n}}{3}^{\mathrm{n}+1}}{\mathrm{n}+1} \\ \mathbf{\Rightarrow }\frac{{\mathbf{4}}^{\mathbf{n}\mathbf{+}\mathbf{1}}\mathbf{-}\mathbf{1}}{\mathbf{n}\mathbf{+}\mathbf{1}}{\mathbf{=}}^{\mathbf{n}}{\mathbf{C}}_{\mathbf{0}}\mathbf{3}\mathbf{+}\frac{{}^{\mathbf{n}}{\mathbf{C}}_{\mathbf{1}}{\mathbf{3}}^{\mathbf{2}}}{\mathbf{2}}\mathbf{+}\frac{{}^{\mathbf{n}}{\mathbf{C}}_{\mathbf{2}}{\mathbf{3}}^{\mathbf{3}}}{\mathbf{3}}\mathbf{+}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{+}\frac{{}^{\mathbf{n}}{\mathbf{C}}_{\mathbf{n}}{\mathbf{3}}^{\mathbf{n}\mathbf{+}\mathbf{1}}}{\mathbf{n}\mathbf{+}\mathbf{1}} \end{gathered}$$

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