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, 11…• Sometimes though, one method is better…

Arithmetic Sequences:

• We know this is Arithmetic because…• 3, 5, 7, 9, 11…

3 5 7 9 11

+ 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 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…

The difference???

ARITHMETIC

• 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

RECURSIVE

• Well, we know d and 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 an if we know an-1

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

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 and a8 = 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.

Continued…

• So we had a8 = a4 + 4d and that a4 = 93 and a8 = 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…

WOW, still not done???

• We know a4 = 93 and a8 = 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

GEOMETRIC SEQUENCES

Let’s look at…

• 1, 5, 25, 125, 625,…• 1 5 25 125 625

x5 x5 x5 x5…so this time, instead of a common

“difference”…We have a common ratio.

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.

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

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 -1 which is 1,953,125.• Yes, geometric sequences can grow rapidly (as

does anything exponential).