In Exercises 7 and 8, write the first five terms of the recursively defined sequence. a1=6, ak+1

In Exercises 7 and 8, write the first five terms of the...

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Key Concepts

Recurrence Relation

A recurrence relation is a mathematical expression that defines each term in a sequence as a function of the preceding term or terms. This relation is the cornerstone of recursively defined sequences, dictating how the sequence evolves from the initial condition.

Iteration

Iteration refers to the process of repeatedly applying the recurrence relation to compute subsequent terms of the sequence. This systematic approach helps in finding specific terms or in understanding the overall behavior of the sequence.

Recursive Sequence

A recursive sequence is a sequence of numbers in which each term is defined based on one or more previous terms using a fixed rule. This method of definition emphasizes the relationship between consecutive terms, allowing the sequence to be generated step-by-step.

Initial Condition

The initial condition of a recursive sequence provides the starting value(s) from which the sequence is built. It is essential because it anchors the sequence and ensures the uniqueness of the sequence generated by the recurrence relation.

Transcript

00:01 Okay, so this question asked to find the first five terms of this infinite sequence.

00:06 First term is given to a1 is equal to six, and then to find out the rest, we're given this formula, which is a subk plus 1 equals 1 3rd, a subk squared.

00:19 So essentially what we're going to do is plug in the last value of the sequence in for ak to find the next one, which would be a.

00:30 Sub k plus 1.

00:32 This will be more clear once we go through the actual problem.

00:36 So we're given a1 already, so the next one we need to find is a2.

00:43 Now, a2 is just a1 plus 1.

00:51 So i'm just substituting in a subk plus 1 to show that what we're using as ak is really just this, which is a1, in 4.

01:06 What this value is going to be a sub k.

01:09 So we'll get one -third quantity.

01:13 Remember, ak is just the last value of the sequence, which would be a1, which we know is 6.

01:21 That value is squared.

01:24 So then we have one -third of 36, which is just 12.

01:31 So a2 is equal to 12.

01:34 Now we'll use a2 to find a3, because remember again, if we're finding a3, we look at this formula right here.

01:47 If we're trying to find a3, then we have to use the previous term of the sequence, which is a2, square that value and then divided by 3.

01:57 So same process again...

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