![]() Arithmetic - a n = a n-1 + d and Geometric - a n = r(a n-1). I modified the lesson to give a little more direction, but not too much! I enjoyed watching the students struggle and then finally seeing the relationship. ![]() The students did an awesome job until they were asked to produce the general recursive formula. From that, I hoped that students would be able to see a pattern! On a note card, I had them answer three scribe any patterns that you see, write a general recursive arithmetic formula, and write a general recursive geometric formula. Under a Creative Commons Attribution-ShareAlike 4.0 License.I created a cut and paste activity where students had to match the sequence, the type, the common difference/ratio, the explicit formula, and the recursive formula. This means that using a recursive formula when using a computer to manipulate a sequence might mean that the calculation will be finished quickly. As can be seen from the examples above, working out and using the previous term a n − 1 can be a much simpler computation than working out a n from scratch using a general formula. ![]() One can work out every term in the series just by knowing previous terms. Recursive equations are extremely powerful. Using this equation, the recursive equation for this geometric sequence is: Suppose each infected person will infect two more people, such that the terms follow a geometric sequence.Įach person infects two more people with the flu virus, making the number of recently-infected people the nth term in a geometric sequence. Depending on how the sequence is being used, either the recursive definition or the non-recursive one might be more useful.Ī recursive geometric sequence follows the formula:Īn applied example of a geometric sequence involves the spread of the flu virus. ![]() In this equation, one can directly calculate the nth-term of the arithmetic sequence without knowing the previous terms. The above equation is an example of a recursive equation since the nth term can only be calculated by considering the previous term in the sequence. When discussing arithmetic sequences, you may have noticed that the difference between two consecutive terms in the sequence could be written in a general way: This definition is valid because, for all n, the recursion eventually reaches the base case of 0.įor example, we can compute 5 ! by realizing that 5 != 5 ⋅ 4 !,Īnd that 4 != 4 ⋅ 3 !, and that 3 != 3 ⋅ 2 !, and that 2 != 2 ⋅ 1 !, and that: 1 != 1 ⋅ 0 ! A recursive definition of a function defines values of the function for some inputs in terms of the values ofįor example, the factorial function n! is defined by the rules: In mathematical logic and computer science, a recursive definition, or inductive definition, is used to define an object in terms of itself. The recursive definition for a geometric sequence is: a n=r⋅a n−1. The recursive definition for an arithmetic sequence is: a n=a n− 1+d.In mathematical logic and computer science, a recursive definition, or inductive definition, is used to define an object in terms of itself.Use a recursive formula to find specific terms of a sequence A recursive definition of a function defines its values for some inputs in terms of the values of the same function for other inputs.
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