𝑇 < 𝑇 , to be greater than 0 but less than 1. Sequence is decreasing, we will expect any multiplier, which is a number of Similarly, to get from the second term 𝑇 to the thirdĬontinuing in this way, we can check the multiplier values for each pair ofĬonsecutive terms in the sequence and see if they agree. In general, to get from the first term 𝑇 to the second To find the value of this constant, we can use the following idea. We can get from one term to the next by multiplying by a constant. Such a pattern suggests that we should instead check if Sequence cannot involve subtracting a constant, because the terms decreaseīy ever smaller amounts. However, the above diagram clearly shows that any recursive formula for the To get from the second term 𝑇 = 1 to the third term To getįrom the first term 𝑇 = 3 to the second term 𝑇 = 1 , we subtract 2. In this case, the terms are decreasing steadily, so we have to subtract. Term can be obtained from the previous one by adding or subtracting a constant. A sensible approach is to check first if each We start by inspecting the sequence to see if we can spot the pattern of Suppose we are asked to find a recursive formula for this sequence: Must work backward from those terms to find a recursive formula. In some questions, we are given the first few terms of a recursive sequence and Note that this is actually an arithmetic sequence with a first term of We deduce that the first five terms of this sequence are Similarly, by substituting 𝑛 = 2, 3, and 4 into the given 𝑛 = 1 into the recursive formula and use the fact that The recursive formula 𝑇 = 𝑇 + 5 . Here, we are given the first term 𝑇 = − 1 3 together with To generate a sequence from its recursive formula, we need to know the first Recall that a recursive formula of the form 𝑇 = 𝑓 ( 𝑇 ) defines each term of a sequence as a function Let us now try an example to practice this skill.Įxample 1: Finding the First Five Terms of a Sequence Using Its Recursive Formulaįind the first five terms of the sequence with general term That we will meet later on in this explainer. Types of sequence, including arithmetic sequences, geometric sequences, and others The strength of recursive formulas is that they enable us to describe many different Note that this is actually anĪrithmetic sequence with a first term of 8 and a common difference of − 4. Therefore, the first five terms of this sequence are Into the formula and use the fact that 𝑇 = − 4 to get To find 𝑇 , we substitute 𝑛 = 3 into theįormula and use the fact that 𝑇 = 0 to getįinally, to find 𝑇 , we substitute 𝑛 = 4 Similarly, to find 𝑇 , we substitute 𝑛 = 2 into the formula and use the fact that
𝑛 = 1 into the recursive formula 𝑇 = 𝑇 − 4 To find the second term, 𝑇 , we substitute Sequence defined by the recursive formula Note that this is actually anĪrithmetic sequence with a first term of 10 and a common difference of 1.Īs another example, suppose we are asked to find the first five terms of the Therefore, the first four terms of this sequence are Into the formula and use the fact that 𝑇 = 1 2 to get Into the formula and use the fact that 𝑇 = 1 1 to getįinally, to find 𝑇 , we substitute 𝑛 = 3 Similarly, to find 𝑇 , we substitute 𝑛 = 2 To find the second term, 𝑇 , we substitute 𝑛 = 1 into the recursive formula
We already know the first term, which is 𝑇 = 1 0 . Suppose we areĪsked to find the first four terms of the sequence defined by the recursive This process is best illustrated through some specific examples. In this way, we canīuild up the sequence until it has as many terms as we wish. The formula with 𝑛 = 2 to derive the value of Once we know the value of 𝑇 , we can use 𝑇 = 𝑓 ( 𝑇 ) , we can use the formula with If we know the first term, 𝑇 , and the recursive formula Of a sequence using a preceding term or terms.Ī recursive formula of the form 𝑇 = 𝑓 ( 𝑇 ) ĭefines each term of a sequence as a function of the previous term. Definition: Recursive Formula of a SequenceĪ recursive formula (sometimes called a recurrence relation) is a formula that
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