<s>[INST] Induction. Am I missing something or is there a mistake in the question? $$\sum_{k=1}^n k*3^k=\frac {3(3^n(2n-1)+1)} 4 $$
So let f(n)= $\sum_{k=1}^n k*3^k $
and  g(n)=$\frac {3(3^n(2n-1)+1)} 4$
By induction hypothesis, $f(n+1) = f(n) + (n+1)3^{n+1} \overset{\text{i.h.}}{=} g(n) + (n+1) 3^{n+1} = g(n+1).$
$$\frac{3(3^{n+1}(2n+1)+1)}4=(n+1)(3^{n+1})+\frac{3(3^n(2n-1)+1))}4 $$
I am stuck afterward, please help, thanks.
 [/INST] To do the inductive step, observe that:
$(n+1)3^{n+1}+\frac{3(3^n(2n-1)+1)}4=\frac{4(n+1)3^{n+1}+3(3^n(2n-1)+1)}4=\frac{(6n+3)3^{n+1}+3}4=\frac{3(2n+1)3^{n+1}+3}4=\frac{3(3^{n+1}(2(n+1)-1)+1)}4$...
</s>