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Suppose I have a business that has expenses totaling 200,000 per month. Currently its revenue is 100,000 per month. The revenue is growing 10% per month.

We can calculate that the business will reach profitability using (log(200000 / 100000) / log(1.1)) = ~7.27 months.

The question is, what is the cash shortfall during those 7.27 months? It feels like a geometric series but I am not sure how to represent the first term.

Finbarr
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The expenses are constant per month, so just multiply to get the total over $7.27$ months. Presumably you are asking for the shortfall to breakeven, but that is not what the question says. Yes, the revenue is a geometric series, so just sum it. You have to decide what to do about the fractional month. You are so close to breakeven you could just ignore it. The shortfall per month is not a geometric series, so you have to look at income and expenses separately.

Ross Millikan
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  • You're right - I edited the question. I'm not looking for the total spent over the period. Rather, I'm looking for the cash shortfall. – Finbarr Apr 13 '20 at 02:02
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    I answered that. The shortfall is neither geometric nor arithmetic. It is the difference of a geometric series from a constant, so sum them both and subtract. – Ross Millikan Apr 13 '20 at 04:35