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RECURSIVE SEQUENCES vs. ARITHMETIC SEQUENCES

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:.

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RECURSIVE SEQUENCES vs. ARITHMETIC SEQUENCES

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  1. RECURSIVE SEQUENCESvs. ARITHMETIC SEQUENCES

  2. 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…

  3. Arithmetic Sequences: • We know this is Arithmetic because… • 3, 5, 7, 9, 11… 3579 11 + 2 + 2 + 2 +2 And 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

  4. RECURSIVE SEQUENCE • We know this is Arithmetic…but 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 nth…or the “n-1”th • Back to 3, 5, 7, 9, 11… • Which we said was written as an= 2n + 1…

  5. The difference??? ARITHMETIC RECURSIVE Well, we know dand a1… a1 = 3 a2 = a1+ d = 3 + 2 = 5 a3= a2+ d = 5 + 2 = 7 a4= a3+ d = 7 + 2 = 9 a5= a4+ d = 9 + 2 = 11 a192= a191+ 2 = …ouch! We can only define anif we know an-1 • an = 2n + 1 • a1= 2(1)+ 1 = 3 • a2= 2(2)+ 1 = 5 • a3= 2(3)+ 1 = 7 • a4= 2(4)+ 1 = 9 • a5= 2(5)+ 1 = 11 • a192 = 2(192) + 1 = 385

  6. The “Formula” • The 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 + d

  7. Recall… • Find the n-th term and the first three terms of the arithmetic sequence having a4 = 93 and a8 = 65. • Yikes! Now what??? • Let’s look at what we have… a4 = 93 anda8 = 65. Also we know an = a + (n – 1)d. • Since a4 and a8 are 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.

  8. Continued… • So we had a8 = a4 + 4d and that a4 = 93 anda8 = 65. • 65 = 93 + 4d. • -28 = 4d • -7 = d • Also, 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 – 21… • …a = 114…

  9. WOW, still not done??? • We know a4 = 93 anda8 = 65 • We found that d = -7 and a = 114 • SOOOO…. • From an = a + (n – 1)d, we get: • an = 114 + (n – 1)(-7) • So the nth term is an = 114 -7n + 7… • an = 121 -7n …and from this we get… • a2 = 107 and a3 = 100

  10. GEOMETRIC SEQUENCES

  11. Let’s look at… • 1, 5, 25, 125, 625,… • 1 5 25 125 625 x5 x5x5x5 …so this time, instead of a common “difference”… We have a common ratio.

  12. Sequences that share a common ratio are said to be… • GEOMETRIC • 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.

  13. The “Formula” • The 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 as… • an = a1*r n -1

  14. Back to… • 1, 5, 25, 125, 625,… • We found the common ratio was 5. • What is the 10th term? • an = a1* r n -1 • a10= 1* 5 10 -1which is 1,953,125. • Yes, geometric sequences can grow rapidly (as does anything exponential).

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