$\text{ The positive integer ' }n\text{ ' for which }2.2^2+3.2^3+4.2^4+..+n.2^n=2^{n+10}\text{ is }$

$\begin{array}{l}{2}^{n+10}={2.2}^2+{3.2}^3+{4.2}^4+..+\mathrm{n}\cdot 2^{\mathrm{n}}\\ \frac{2\left(2^{n+10}\right)={2.2}^3+{3.2}^4+..+(\mathrm{n}-1)2^n+\mathrm{n}\cdot 2^{n+1}}{-2^{\mathrm{n}+10}=2\cdot 2^2+2^3+2^4+..+2^{\mathrm{n}}-\mathrm{n}\cdot 2^{\mathrm{n}+1}}\\ =8+8\left(2^{2\mathrm{n}-2}-1\right)-\mathrm{n}\cdot 2^{n+1}\\ =8+2^{n+1}-8-\mathrm{n}\cdot 2^{n+1}=2^{n+1}-\mathrm{n}\cdot 2^{n+1}\\ \mathrm{n}=513\end{array}$