Step 1: Write the node equation at the summing junction.
In positive feedback, the feedback signal is added to the input. Let $V_e$ be the effective input to the amplifier. Then $V_e = V_i + \beta V_o$, where $\beta V_o$ is the portion of output fed back and added (not subtracted) to the input.
Step 2: Express output in terms of $V_e$.
The amplifier multiplies its input by gain $A$: $V_o = A \cdot V_e = A(V_i + \beta V_o)$. Expanding: $V_o = A V_i + A\beta V_o$.
Step 3: Solve for closed-loop gain.
Rearranging: $V_o - A\beta V_o = A V_i \Rightarrow V_o(1 - A\beta) = A V_i$. Therefore the closed-loop gain is: \[ \boxed{\dfrac{A}{1 - A\beta}} \]