A recursive rule gives the beginning term or terms of a sequence, and a recursive equation tells how a_n is related to one or more preceding terms.
See solution.
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To define a sequence recursively, we need a recursive rule. This type of rule gives the beginning term or terms of a sequence, and a recursive equation tells how the nth term, a_n, is related to one or more preceding terms. For example, let's choose two initial terms.
Example initial terms
a_1=1 and a_2=2Now, we just need to define a relation that tells us how to obtain the next term from these initial terms chosen. By writing this relation algebraically we obtain the sequence's recursive equation. The relation can be any. For example, the next term is the product of the two previous terms.
Recurisive Equation
a_n = (a_(n-1))(a_(n-2))
For this example, this recursive rule defines the recursive sequence shown below.
1, 2, 1* 2 ⟶ 2, 2*2 ⟶ 4, 2* 4 ⟶ 8, 4 * 8 ⟶ 32, ...
