jmc

Let $$S=2\cdot 2^2+3\cdot 2^3+4\cdot 2^4+\cdots +n\cdot 2^n.$$Then $$2S=2\cdot 2^3+3\cdot 2^4+4\cdot 2^5+\cdots +n\cdot 2^{n+1}.$$Subtracting these equations, we get $$\begin{array}{rl}S& =(2\cdot 2^3+3\cdot 2^4+4\cdot 2^5+\cdots +n\cdot 2^{n+1})-(2\cdot 2^2+3\cdot 2^3+4\cdot 2^4+\cdots +n\cdot 2^n)\\ & =-2\cdot 2^2-2^3-2^4-\cdots -2^n+n\cdot 2^{n+1}\\ & =-8-2^3(1+2+2^2+\cdots +2^{n-3})+n\cdot 2^{n+1}\\ & =-8-2^3(2^{n-2}-1)+n\cdot 2^{n+1}\\ & =-8-2^{n+1}+8+n\cdot 2^{n+1}\\ & =(n-1)2^{n+1}.\end{array}$$Hence, $(n-1)2^{n+1}=2^{n+10},$ so $n-1=2^9=512,$ from which $n=\boxed{513}.$