<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] You can calculate $$\frac{3^{n+2}(2n+1)-3-3^{n+1}(2n-1)-3}{4}=\frac{3^{n+1}(3(2n+1)-2n+1)}{4}=\frac{3^{n+1}(6n+3-2n+1)}{4}=\frac{3^{n+1}(6n+3-2n+1)}{4}=…$$
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