![]() ![]() Since our recursion uses the two previous terms, our recursive formulas must specify the first two terms. ![]() It turns out that each term is the product of the two previous terms. Solution The terms of this sequence are getting large very quickly, which suggests that we may be using either multiplication or exponents. Since our recursion involves two previous terms, we need to specify the value of the first two terms:Įxample 4: Write recursive equations for the sequence 2, 3, 6, 18, 108, 1944, 209952. Each term is the sum of the two previous terms. Solution: This sequence is called the Fibonacci Sequence. Solution: The first term is 2, and each term after that is twice the previous term, so the equations are:Įxample 3: Write recursive equations for the sequence 1, 1, 2, 3, 5, 8, 13. Notice that we had to specify n > 1, because if n = 1, there is no previous term!Įxample 2: Write recursive equations for the sequence 2, 4, 8, 16. Solution: The first term of the sequence is 5, and each term is 2 more than the previous term, so our equations are: Recursive equations usually come in pairs: the first equation tells us what the first term is, and the second equation tells us how to get the n th term in relation to the previous term (or terms).Įxample 1: Write recursive equations for the sequence 5, 7, 9, 11. For the following exercises, use the recursive formula to write the first five terms of the arithmetic sequence. If a sequence is recursive, we can write recursive equations for the sequence. In a geometric sequence, each term is obtained by multiplying the previous term by a specific number. What this shows is that a recurrence can have infinitely many solutions. Note that s n 17 2 n and s n 13 2 n are also solutions to Recurrence 2.2.1. Thus a solution to Recurrence 2.2.1 is the sequence given by s n 2 n. ![]() Why? In an arithmetic sequence, each term is obtained by adding a specific number to the previous term. A solution to a recurrence relation is a sequence that satisfies the recurrence relation. Create a recursive formula by stating the first term, and then stating the formula to be the previous term plus the common difference. If we go with that definition of a recursive sequence, then both arithmetic sequences and geometric sequences are also recursive. Recursion is the process of starting with an element and performing a specific process to obtain the next term. Pay attention to whether the 1 is being added or subtracted to decide which term the notation is referring to.We've looked at both arithmetic sequences and geometric sequences let's wrap things up by exploring recursive sequences. If a_1 is the first term, the successive terms of the geometric sequence follow this same pattern. The first term of the sequence should always be defined, and is often a_1. Since sequence notation looks similar to other types of mathematical notation, such as exponential notation, it can be easy to confuse them. This means that even though the sequence is showing negative integers rather than positive integers, it is still increasing. This sequence has a constant difference of +8. But not necessarily if the terms are negative. Confusing geometric and arithmetic sequences An arithmetic sequence has a common difference from term to term. ![]() If the common difference is negative, this is true.
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