I want to change the oil in my auto transmission. It holds 16 quarts. I can only remove 4 quarts at a time. If I remove 4 quarts of existing oil and then add 4 quarts of new oil, the existing mixture will be improved. Is there an math expression that will tell me at what rate the transmission oil will improve as I perform this 4/4 exchange?

We have to assume that the new and old oil is completely mixed between each 4-quart change (you'll have to run the engine and drive around the block a few times between to make sure of this). With each change the fraction of old oil left is reduced by 1/4, so the series for old oil concentration is 1, 3/4, (3/4)^2, (3/4)^3, etc. Subtract these values from 1 to find the fraction of new oil after each change..