What Is A Sequence

Formally, a sequence is one enumerated arsenal of objects, but informally, a succession is a countable framework representing one ordered perform of elements or numbers.

You are watching: Which recursively defined function represents the sequence


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Definition Sequence


And us specify a sequence either recursively or explicitly.

Recursive Formula Definition

So, what is recursion?

A recursive definition, sometimes dubbed an inductive definition, is composed of two parts:

Recurrence RelationInitial Condition

A recurrence relation is an equation that uses a ascendancy to generate the next term in the sequence from the previous ax or terms. In other words, a recurrence relationship is one equation that is identified in terms of itself.

And every recurrence relations must come through an initial condition, i m sorry is a perform of one or more terms of the sequence that precede the an initial term whereby the recurrence relationship starts.

Example

For instance,


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List state Recursive — Example


Notice that this looks as with the procedure we usage for math induction!

The idea behind inductive proofs is comparable to a staircase, together the only means to reach the peak is to rise all the steps before it, as noted by math Bits. The same thing is happening through recursion – each action is produced from the step or steps preceding.


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Staircase Analogy


Recursive Formulas because that Sequences

Alright, so as we’ve just noted, a recursive succession is a succession in which state are identified using one or more previous terms along with an early condition. And also the most standard recursive formula is the Fibonacci sequence.

The Fibonacci sequence is as follows: 0, 1, 1, 2, 3, 5, 8, 13, 21,…

Notice the each number in the sequence is the amount of the 2 numbers the precede it. Because that example, 13 is the sum of 5 and also 8 which are the two coming before terms.

In fact, the flower of a sunflower, the shape of galaxies and hurricanes, the kinds of leaves on tree stems, and also even molecular DNA every follow the Fibonacci sequence which when each number in the succession is drawn as a rectangular width creates a spiral.

Isn’t it remarkable to think that math have the right to be it was observed all approximately us?

But, sometimes using a recursive formula can be a bit tedious, together we continually have to rely top top the preceding terms in stimulate to generate the next.

So now, let’s revolve our attention to defining sequence clearly or generally. Every this way is the each term in the sequence deserve to be calculated directly, without knowing the ahead term’s value.

Example

In this problem,

What we will an alert is the patterns begin to pop-up together we create out regards to our sequences. And it’s in these trends that we can uncover the nature of recursively defined and explicitly defined sequences.

We want to repeat ourselves of some necessary sequences and also summations from Precalculus, such together Arithmetic and also Geometric sequences and also series, that will aid us uncover these patterns.


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Arithmetic succession Formula


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Geometric sequence Formula


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Summation sequence Formulas


Armed through these summation formulas and also techniques, we will begin to generate recursive formulas and closed formulas for other sequences with similar patterns and also structures.

Example

So, utilizing our recognized sequences, let’s find a recursive meaning for the succession 4,9,14,19,24,29,…


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Arithmetic Recursive — Example


Now, making use of our known summation formulae, let’s uncover a closed definition for the same sequence that 4,9,14,19,24,29,…


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Closed type Arithmetic Sequence


Additionally, we will find a superb procedure for finding the amount of an Arithmetic and also Geometric sequence, using Gauss’s exploration of reverse-add and multiply-shift-subtract, respectively.

Example

Suppose we want to uncover the sum of the adhering to sequence: 1,3,5,7,9,..,39.

First, we require to find the close up door formula because that this arithmetic sequence. To do this, we need to identify the common difference which is the amount the is being included to each term that will generate the following term in the sequence. The easiest way to find it is come subtract two nearby terms. So, because that our current example, if we subtract any two nearby terms us will notice that the common difference is 2.


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Find Closed type Of Arithmetic Sequence


Now, we deserve to use the clearly formula to determine the variety of terms the we space summing.

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Determine The number of Terms In Arithmetic Series


Finally, we apply the reverse and also add technique to discover the sum, where we very first list every the terms in one direction, climate reverse and list every the terms in opposing direction. In other words, we will certainly “wrap” the series back top top itself, together MathBitsNotebook unique states.


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Gauss method Sum


Summary

Throughout this video, we will see exactly how a recursive formula calculates each term based on the ahead term’s value, so it takes a bit much more effort to create the sequence. In contrast, an clear formula straight calculates each term in the sequence and also quickly finds a specific term.

Both formulas, together with summation techniques, are invaluable come the examine of counting and recurrence relations. And with these brand-new methods, we will certainly not only have the ability to develop recursive formulas for details sequences, but we will be on our way to solving recurrence relations!

So, let’s jump right in and also discover the fun!

Video indict w/ full Lesson & detailed Examples

1 hr 49 min

Introduction to Video: Recursive Formula — order — Summations00:00:51 can you guess: v the pattern and determine the following term in the sequence? (Examples #1-7)00:11:37 What is a Recursive an interpretation and clear Formula?00:21:43 uncover the an initial five terms of the succession (Examples #8-10)00:30:38 Recursive formula and closed formula for Arithmetic and Geometric Sequences00:40:27 triangular — Square — Cube — Exponential — Factorial — Fibonacci Sequences00:47:42 uncover a recursive definition for every sequence (Examples #11-14)01:00:11 Use recognized sequences to find a closed formula (Examples #15-20)01:22:29 using reverse—add an approach on Arithmetic sequences (Examples #21-22)01:35:48 Summing Geometric Sequences using multiply—shift—subtract method (Examples #23-34)01:44:00 Summation and Product Notation (Example #25a-d)Practice Problems with Step-by-Step solutions Chapter Tests with video clip Solutions