# RECURSIVE SEQUENCES vs. ARITHMETIC SEQUENCES

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RECURSIVE SEQUENCES vs. ARITHMETIC SEQUENCES. Sequences. As you may have discovered, there are times when an ARITHMETIC sequence can be written as a RECURSIVE sequence. Look at 3 , 5, 7, 9, 11… Sometimes though, one method is better…. Arithmetic Sequences:. - PowerPoint PPT Presentation

### Text of RECURSIVE SEQUENCES vs. ARITHMETIC SEQUENCES

SEQUENCES

RECURSIVE SEQUENCESvs. ARITHMETIC SEQUENCES

Sequences As you may have discovered, there are times when an ARITHMETIC sequence can be written as a RECURSIVE sequence.Look at 3, 5, 7, 9, 11Sometimes though, one method is better

Arithmetic Sequences:We know this is Arithmetic because3, 5, 7, 9, 11 357911

+ 2 + 2 + 2 +2And from an = a + (n 1)d, we know the nth term would be an = a + (n 1)d an = 3 + (n 1)2 or an = 3 + 2n 2, which becomes an = 2n + 1

RECURSIVE SEQUENCEWe know this is Arithmeticbut is it also Recursive? What is Recursive? To be Recursive means that the nth term in a sequence is defined by the term before the nthor the n-1th Back to 3, 5, 7, 9, 11 Which we said was written as an = 2n + 1

The difference???ARITHMETICan = 2n + 1a1 = 2(1) + 1 = 3a2 = 2(2) + 1 = 5a3 = 2(3) + 1 = 7a4 = 2(4) + 1 = 9a5 = 2(5) + 1 = 11a192 = 2(192) + 1 = 385

RECURSIVEWell, we know d and a1 a1 = 3a2 = a1 + d = 3 + 2 = 5a3 = a2 + d = 5 + 2 = 7a4 = a3 + d = 7 + 2 = 9a5 = a4 + d = 9 + 2 = 11a192 = a191 + 2 = ouch!We can only define an if we know an-1The FormulaThe term which you seek is called an.The common difference between any two terms is called d.The term before the term you seek is called an-1.Since you get the next term by adding the common difference, the value of a2 is just a1 + d. The third term is a3 = (a1 + d) + d = a2 + d. The fourth term is a4 = (a2 + d) + d = a3 + d. Following this pattern, the n-th term an will have the form an = an-1 + dRecallFind the n-th term and the first three terms of the arithmetic sequence having a4 = 93 and a8 = 65.Yikes! Now what??? Lets look at what we have a4 = 93 and a8 = 65. Also we know an = a + (n 1)d.Since a4and a8are four places apart, then I know from the definition of an arithmetic sequence that a8= a4+ 4d. Using this, I can then solve for the common difference d.

ContinuedSo we had a8= a4+ 4d and that a4 = 93 and a8 = 65.65 = 93 + 4d.-28 = 4d-7 = dAlso, I know that a4 = a + (4 1)d, so, using the value I just found for d, I can find the value of the first term a.So 93 = a + (4-1)(-7) 93 = a 21a = 114WOW, still not done???We know a4 = 93 and a8 = 65We found that d = -7 and a = 114SOOOO.From an = a + (n 1)d, we get:an = 114 + (n 1)(-7)So the nth term is an = 114 -7n + 7an = 121 -7n and from this we geta2 = 107 and a3 = 100GEOMETRIC SEQUENCES

Lets look at1, 5, 25, 125, 625,1 525125625x5 x5 x5 x5so this time, instead of a common differenceWe have a common ratio.

Sequences that share a common ratio are said to beGEOMETRIC KEY TERMS:A sequence is a set of numbers in a specific order.The numbers in the sequence are called terms.The difference between the terms is called the common ratio.If the difference between successive terms has a pattern that can be defined as a ratio, then it is called an geometric sequence.This sequence can either be a set interval or repeating.The FormulaThe term which you seek is called an.The common ratio between any two terms is called r.Since you get the next term by multiplying the first term by this ratio, then a geometric sequence can be defined asan = a1* r n -1Back to1, 5, 25, 125, 625,We found the common ratio was 5.What is the 10th term?an = a1* r n -1a10 = 1* 5 10 -1 which is 1,953,125.Yes, geometric sequences can grow rapidly (as does anything exponential). ##### On Polynomial Recursive Sequences - Dagstuhl ... 117:6 OnPolynomialRecursiveSequences Poly-recursive sequences. We now generalise the concept of linear recursive sequences byallowing
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