# Research Math Proofs

Back to Research Math.

We begin with a basic mathematical description of the rate of technology gain in a country, denoted here as the TechProgress term: We consider TechProgress here as a function of the size of a country (CountrySize). Thus: We will proceed from this relation.

Now observe the impact of adding x provinces to CountrySize: Expand the denominator as per the Technology Cost formula to make the relation to CountrySize explicit. And expand the TotalResearchIncome[CountrySize + x] term:  So we have: The rest is algebraic manipulation...      And, thus, we have: Research Rate Change

To derive the equality condition, take "Change in Research Rate" = 0, and add 1 to both sides: Continue manipulating:    Now, note that  Substitute these expressions to get: And, finally: Equality Condition

This simplifies for large (CountrySize ≥ 8) countries to:   Therefore: Equality Condition for Large Countries

Note that in both the Equality Condition and Research Rate Change formulae, the marginal cases may be easily derived by taking x = 1, and making the substitution AverageProvinceResearchIncome = NewProvinceResearchIncome.