![]() ![]() To begin, we need to define how we’re going to measure n. Plotting the values we calculated for Marco’s collection, we can see the values form a straight line, the shape of linear growth. This constant change is the defining characteristic of linear growth. In this example, Marco’s collection grew by the same number of bottles every year. The steps of determining the formula and solving the problem of Marco’s bottle collection are explained in detail in the following videos. So Marco will reach 1000 bottles in 18 years. We can now also solve for when the collection will reach 1000 bottles by substituting in 1000 for P n and solving for n Using this equation, we can calculate how many bottles he’ll have after 5 years: You can probably see the pattern now, and generalize that We can do so by selectively not simplifying as we go: While you may already be able to guess the explicit equation, let us derive it from the recursive formula. An explicit equation allows us to calculate P n directly, without needing to know P n-1. For that, a closed or explicit form for the relationship is preferred. While recursive relationships are excellent for describing simply and cleanly how a quantity is changing, they are not convenient for making predictions or solving problems that stretch far into the future. However, solving how long it will take for his collection to reach 1000 bottles would require a lot more calculations. We have answered the question of how many bottles Marco will have in 5 years. Using this relationship, we could calculate: ![]() Here, the number of bottles in year n can be found by adding 32 to the number of bottles in the previous year, P n-1. A recursive relationship is a formula which relates the next value in a sequence to the previous values. We could describe how Marco’s bottle collection is changing using: So P 0 would represent the number of bottles now, P 1 would represent the number of bottles after 1 year, P 2 would represent the number of bottles after 2 years, and so on. ![]() Suppose that P n represents the number, or population, of bottles Marco has after n years. While you could probably solve both of these questions without an equation or formal mathematics, we are going to formalize our approach to this problem to provide a means to answer more complicated questions. ![]()
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