Induction without a base case [duplicate]

I am looking for an example where you have $P(n)$ implying $P(n+1)$. However there is no base case. For which there is therefore no solution at all for the induction problem even though the inductive step itself works.

Suppose we want to prove $n=n+1$ for all (positive) integers $n$. We omit the base case. The induction hypothesis is $k=k+1$ for some $k\in \mathbb N$. Adding $1$ to both sides gives $k+1=k+1+1$, or $(k+1)=(k+1)+1$, which is the statement to be proven for $n=k+1$. Thus, we have completed the induction step, but there is no base for which this is true, so the statement won't have to be true too.

$$1+2+3+\dots+n=\frac{n(n+1)}2+\pi$$ $$1+2+4+\dots+2^{n-1}=2^n$$