summation - Find the sum of the n terms of the series $2cdot2^0+3cdot2^1+4cdot2^2+dots$

Find the sum of the n terms of the series:

$2\cdot2^0+3\cdot2^1+4\cdot2^2+\dots$

I don't know how to proceed. Please explain the process and comment on technique to solve questions of similar type.

Source: Barnard and Child Higher Algebra.

Thanks in Advance!

Answer

The formula for the sum of a geometric sequence gives

$$
\sum_{k=0}^n2^kx^{k+2}=\frac{x^2-2^{n+1}x^{n+3}}{1-2x}\tag{1}
$$
Differentiating $(1)$ yields
$$
\sum_{k=0}^n(k+2)2^kx^{k+1}=\frac{2x-(n+3)2^{n+1}x^{n+2}}{1-2x}+\frac{2x^2-2^{n+2}x^{n+3}}{(1-2x)^2}\tag{2}
$$
Plugging in $x=1$ leads to
$$
\sum_{k=0}^n(k+2)2^k=(n+1)2^{n+1}\tag{3}

$$