$\text{ Given }\alpha \text{ and }\beta \text{ are the roots of the quadratic equation }{x}^2-4x+k=0(k\ne 0).\text{ If }$$\alpha \beta ,\alpha {\beta }^2+{\alpha }^2\beta ,{\alpha }^3+{\beta }^3\text{ are in geometric progression, then the value of }k\text{ is }$

$\begin{array}{l}\text{ Given }\alpha ,\beta \text{ are roots of }{x}^2-4x+k=0(k\ne 0)\\ \Rightarrow \alpha +\beta =4,\alpha \beta =k\\ \text{ Given that }\alpha \beta ,\alpha \beta (\alpha +\beta ),{\alpha }^3+{\beta }^3\text{ are in G.P }\\ \Rightarrow \left(\alpha \beta \right(\alpha +\beta ){)}^2=\alpha \beta \left({\alpha }^3+{\beta }^3\right)\end{array}$

$\begin{array}{l}\Rightarrow \left(\alpha \beta {)}^2\right(\alpha +\beta {)}^2=\alpha \beta (\alpha +\beta )\left({\alpha }^2+{\beta }^2-\alpha \beta \right)\\ \Rightarrow \left(\alpha \beta \right)(\alpha +\beta )=\left({\alpha }^2+{\beta }^2-\alpha \beta \right)\\ \Rightarrow \left(\alpha \beta \right)(\alpha +\beta )=(\alpha +\beta {)}^2-2\alpha \beta -\alpha \beta \\ \Rightarrow \left(\alpha \beta \right)(\alpha +\beta )=(\alpha +\beta {)}^2-3\alpha \beta \\ \Rightarrow \left(k\right)\left(4\right)=(4{)}^2-3k\\ \Rightarrow 7k=16\\ \Rightarrow k=\frac{16}{7}\cong 2.29\end{array}$