$\bf1$. Context:
Payback Period (or PBP): Time required to recover initial investment in a project.
Payback Period = $\bf3.25$ years (to recover the $50,000$)
$\bf2$. Unliked Formula: PBP = $\frac{\text{Cost of Investment}}{\text{Average Annual Cashflow}}$
Reason: Formula assumes uniform cashflows for all the years unlike our example.
$\bf 3$. Question/Problem:
How to write the formula for PBP so that it looks neat and works for non-unifrom cashflows?
$\bf 4$. My take:
I came up with PBP = $n |\left(\sum(\text{CashInflow})-\sum(\text{CashOutflow})=0\right)$. But this makes use of "such that" and also doesn't capture the $0.25$ year thing.