# 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:

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:

This simplifies for large (*CountrySize* ≥ 8) countries to:

Therefore:

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*.

**Research:**

Research (Advanced) • Research Math Proofs • Research strategy • Research Cost - Supporting Data ( and )

**Technology:**

Technology tree • Technology groups • Westernisation • Country size modifiers