<s>[INST] Elementary problems Find $n$ such that $n$ is a positive integer satisfying the following equation.
$$2(2^2)+3(2^3)+4(2^4)+\ldots+n(2^n)=2^{n+10}$$
Can anybody help ? I can't believe this is an elementary school problem...
 [/INST] $\begin{eqnarray} 
{\bf Hint} \  
  && \color{#0a0}0\cdot 2^2\! &+& \color{#c00}2\cdot 2^2\! &+& 3\cdot 2^3\! &+& 4\cdot 2^4\! &+&\, \cdots &\!+& n\cdot 2^n \\
  &&           &=& \color{#0a0}1\cdot 2^3\! &+& \color{#c00}3\cdot 2^3\! &+& 4\cdot 2^4\! &+&\, \cdots &\!+& n\cdot 2^n \\   
  &&           &&            &=& \color{#0a0}2\cdot 2^4\! &+& \color{#c00}4\cdot 2^4\! &+&\,  \cdots &\!+& n\cdot 2^n \\
  &&           &&            &&               &&   \ddots    &&\ \ \, \vdots &&\\   
  &&           &&            &&               &&             &&\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! = \color{#0a0}{(n\!-\!2)}\, 2^n\!\! &+& \color{#c00} n\cdot 2^n\\
  &&           &&            &&               &&            &&  &&\!\!\!\!\!\!\!\!\!\!\!\!\!\! =\color{#0a0}{(n\!-\!1)}\, 2^{n+1} 
\end{eqnarray}$
</s>