![]() 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. This constant is called the Common Difference. An Arithmetic Sequence is such that each term is obtained by adding a constant to the preceding term. Each term is the sum of the two previous terms. A Recursive equation is a formula that enables us to use known terms in the sequence to determine other terms. Solution: This sequence is called the Fibonacci Sequence. 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. 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. A solution to a recurrence relation is a sequence that satisfies the recurrence relation. 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. 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. ![]() Why? In an arithmetic sequence, each term is obtained by adding a specific number to the previous term. 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. nth term of Geometric Progression an an 1 × r for n 2. They are, nth term of Arithmetic Progression an an 1 + d for n 2. There are few recursive formulas to find the nth term based on the pattern of the given data. But not necessarily if the terms are negative. Pattern rule to get any term from its previous terms. If the common difference is negative, this is true. Thinking arithmetic sequences with negative terms always decrease.Always check all terms before deciding the rule. Since this sequence is arithmetic, the rule from term to term is +2. For this reason, always look for the common difference of an arithmetic sequence, instead of using multiplication.Īlthough 2 \times 2=4, this does not work for the rest of the terms. The relationship from term to term in an arithmetic sequence is always additive, not multiplicative. For the following exercises, use the recursive formula to write the first five terms of the arithmetic sequence. And because, the constant factor is called the common ratio20. Multiplying the value for a term to get another term of an arithmetic sequence A geometric sequence18, or geometric progression19, is a sequence of numbers where each successive number is the product of the previous number and some constant. ![]()
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