A sequence is defined recursively. Write down the first five terms. a1=3 ; an=(an-1)/(n)

Name: Problem 41, Chapter 11: Precalculus: Concepts Through Functions, A Unit Circle Approach to Trigonometry
Uploaded: 2020-06-21T17:35:34-08:00
Duration: 2 min 17 s
Channel: Julie Silva
Description: A sequence is defined recursively. Write down the first five terms. a1=3 ; an=(an-1)/(n)

A sequence is defined recursively. Write down the first five...

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Key Concepts

Recurrence Relation

A recurrence relation is a formula that expresses each term of the sequence in relation to one or more of its predecessors. It is the defining rule that generates new terms from already known values and serves as a systematic process for building the entire sequence.

Iterative Computation

Iterative computation refers to the process of repeatedly applying the recurrence relation, using the results of previous steps to calculate further values. This step-by-step method is essential for constructing the terms of the sequence and analyzing its behavior.

Recursive Sequence

A recursive sequence is defined by specifying one or more initial terms and a recurrence relation that determines subsequent terms based on previous terms. This method of defining sequences emphasizes the dependency of each term on those that come before it, and it is a fundamental concept in many areas of mathematics, including algebra and computer science.

Initial Condition

The initial condition in a recursive sequence provides the starting point necessary for generating the sequence. It is a specific value or set of values that uniquely determines the sequence when combined with the recurrence relation, and without it, the remaining terms would be undefined.

Transcript

00:01 We're being asked to find the first five terms in the given recursively defined sequence.

00:05 Well, they tell us that a sub 1 is equal to 3, so we know the first term in the sequence is 3.

00:10 Now, to find the second term in the sequence, we're going to substitute 2 in place of n.

00:14 So we're going to have a sub 2 minus 1, which is 1, divided by 2.

00:18 Well, we know that a sub 1 is equal to 3.

00:20 So our second term is 3 over 2.

00:24 Now, to find the third term in the sequence, we substitute 3 in place of n.

00:28 So we'll have a sub, while 3 minus 1 is 2, all divided by three.

00:33 Well, we know that a.

00:35 2 is equal to three halves.

00:36 So we're going to have three halves divided by three.

00:39 So now we just need to divide.

00:41 Well, three halves.

00:45 Hold on here.

00:45 Let's make sure that's legible.

00:48 So we're going to have three halves divided by three.

00:51 And to divide fractions, we multiply by a reciprocal.

00:53 So we're going to have three halves times one -third, which is equal to one -half because our threes will cancel.

00:59 So the third term in the sequence is one -half.

01:02 Now, to find the fourth term in this sequence, we're going to substitute 4 in place of n...

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