<s>[INST] Finding the sum of $1 \cdot 2x + 2 \cdot 3x^{2} + 3 \cdot 4x^{3} + \dots$ . What is the sum of $1 \cdot 2x + 2 \cdot 3x^{2} + 3 \cdot 4x^{3} + \dots$ ?
I have been stuck on this problem with no direction. I have tried multiplying the sequence with $x$ and trying out $S-Sx$ but have gotten nowhere. Any help?
Thanks.
 [/INST] Hint:
$$S=2x+6x^2+12x^3+20x^4+\cdots$$
$$S(1-x)=2x+(6-2)x^2+(12-6)x^3+(20-12)x^4+\cdots=V\text{(say)}$$
$$V(1-x)=2x+x^2(4-2)+x^3(6-4)+x^4(8-6)+\cdots=\dfrac{2x}{1-x}\iff|x|<1$$
Can you take it from here?
See : this 
</s>