I'm confused a little about not proving base case in strong induction. Can someone point out the error in following proof( I'm using strong induction)?
Theorem: Prove that for all $n \in N$, $1.3^0 + 3.3^1 + 5.3^2 + .. + (2n+1)3^n = n3^{n+1}$
Proof:
Let n be arbitrary natural number. Suppose, for all natural numbers k smaller than n, $1.3^0 + 3.3^1 + 5.3^2 + .. + (2k+1)3^k = k3^{k+1}$
Since (n-1) < n, Then it follows from our assumption that $1.3^0 + 3.3^1 + 5.3^2 + .. (2n-1)3^{n-1} = (n-1)3^n$
So, $1.3^0 + 3.3^1 + 5.3^2 + .. + (2n+1)3^n$ = $1.3^0 + 3.3^1 + 5.3^2 + .. (2n-1)3^{n-1} + (2n+1)3^n$ = $(1.3^0 + 3.3^1 + 5.3^2 + .. (2n-1)3^{n-1}) + (2n+1)3^n$ = $(n-1)3^n + (2n+1)3^n$ = $3n.3^n$ = $n3^{n+1}$
Hence, by the assumptions of strong induction, $1.3^0 + 3.3^1 + 5.3^2 + .. + (2n+1)3^n = n3^{n+1}$