Sequences Sequences and Seriesand Series

It’s all in Section 9.4a!!!It’s all in Section 9.4a!!!

Sequence – an ordered progression of numbers – examples:

1. 5,10,15,20,25

2. 2,4,8,16,32, , 2 ,k

3.1: 1,2,3,kk

4. 1 2 3, , , , ,ka a a a which is sometimes abbreviated ka

Finite SequenceFinite Sequence

Infinite SequencesInfinite Sequences

Notice: In sequence (2) and (3), we are able to define a rule forthe k-th number in the sequence (called the k-th term).

(unless otherwise(unless otherwisespecified, the wordspecified, the word

““sequence” will refer tosequence” will refer toan infinite sequence)an infinite sequence)

Practice ProblemsPractice ProblemsFind the first 6 terms and the 100th term of the sequence

in which ka2 1.ka k

Note: This is an explicit rule for the k-th term2

1 1 1 0a 2

2 2 1 3a 2

3 3 1 8a

24 4 1 15a

25 5 1 24a

26 6 1 35a

2100 100 1 9999a

Practice ProblemsPractice Problems

Find the first 6 terms and the 100th term for the sequence definedrecursively by the following conditions:

1 2n nb b 1 3b

Another way to define sequences is recursively, where wefind each term by relating it to the previous term.

for all n > 1.

1 3b

2 1 2 5b b

3 2 2 7b b

3,5,7,9,11,13,

The pattern???

The sequence:

100 3 99 2 201b

Definition:Definition: Arithmetic Sequence Arithmetic SequenceA sequence is an arithmetic sequence if it can be writtenin the form

ka

for some constant d. , , 2 , , 1 ,a a d a d a n d

The number d is called the common difference.

Each term in an arithmetic sequence can be obtained recursivelyfrom its preceding term by adding d:

(for all n > 2).1n na a d

Practice ProblemsPractice ProblemsFor each of the following arithmetic sequences, find (a) thecommon difference, (b) the tenth term, (c) a recursive rule for then-th term, and (d) an explicit rule for the n-th term.

1. 6, 2,2,6,10, (a) The difference ( d ) between successive terms is 4.

(b) 10 6 10 1 4 30a

(c) 1 6,a 1 4,n na a 2n

(d) 6 1 4 4 10na n n

Practice ProblemsPractice ProblemsFor each of the following arithmetic sequences, find (a) thecommon difference, (b) the tenth term, (c) a recursive rule for then-th term, and (d) an explicit rule for the n-th term.

2. 11,8,5,2, 1, (a) The difference ( d ) between successive terms is –3.

(b) 10 11 10 1 3 16a

(c) 1 11,a 1 3,n na a 2n

(d) 11 1 3 3 14na n n

Practice ProblemsPractice ProblemsFor each of the following arithmetic sequences, find (a) thecommon difference, (b) the tenth term, (c) a recursive rule for then-th term, and (d) an explicit rule for the n-th term.

3. ln 3, ln 6, ln12, ln 24Is this sequence truly arithmetic???

Difference between successive terms:

ln 6 ln 3 ln 6 3 ln 2 ln12 ln 6 ln 12 6 ln 2 ln 24 ln12 ln 24 12 ln 2

We do have acommon difference!!!

Practice ProblemsPractice ProblemsFor each of the following arithmetic sequences, find (a) thecommon difference, (b) the tenth term, (c) a recursive rule for then-th term, and (d) an explicit rule for the n-th term.

(a) The difference ( d ) between successive terms is ln(2).

(b) 10 ln 3 10 1 ln 2a ln 3 9ln 2

9ln 3 2 ln1536

3. ln 3, ln 6, ln12, ln 24

Practice ProblemsPractice ProblemsFor each of the following arithmetic sequences, find (a) thecommon difference, (b) the tenth term, (c) a recursive rule for then-th term, and (d) an explicit rule for the n-th term.

(c)1 ln 3,a 1 ln 2,n na a 2n

(d) ln 3 1 ln 2na n 1ln 3 ln 2n

1ln 3 2n

3. ln 3, ln 6, ln12, ln 24

Geometric Sequences

Definition:Geometric Sequence

A sequence is a geometric sequence if it can be written inthe form

na

for some non-zeroconstant r. 2 1, , , , ,na a r a r a r

The number r is called the common ratio.

Each term in a geometric sequence can be obtained recursivelyfrom its preceding term by multiplying by r :

(for all n > 2).1n na a r

Guided PracticeFor each of the following geometric sequences, find (a) thecommon ratio, (b) the tenth term, (c) a recursive rule for the n-thterm, and (d) an explicit rule for the n-th term.

3,6,12,24,48,1.

(a) The ratio ( r ) between successive terms is 2.10 1

10 3 2a (b)93 2 1536

1 3,a (c) 12 ,n na a 2n13 2nna

(d)

Guided PracticeFor each of the following geometric sequences, find (a) thecommon ratio, (b) the tenth term, (c) a recursive rule for the n-thterm, and (d) an explicit rule for the n-th term.

3 1 1 3 510 ,10 ,10 ,10 ,10 , 2.

(a) Apply a law of exponents:

1

1 3 23

1010 10

10

10 13 210 10 10a

(b)3 18 1510 10

Guided PracticeFor each of the following geometric sequences, find (a) thecommon ratio, (b) the tenth term, (c) a recursive rule for the n-thterm, and (d) an explicit rule for the n-th term.

3 1 1 3 510 ,10 ,10 ,10 ,10 , 2.

31 10 ,a (c)

2110 ,n na a 2n

13 210 10n

na(d)

3 2 210 n 2 510 n

Guided PracticeFor each of the following geometric sequences, find (a) thecommon ratio, (b) the tenth term, (c) a recursive rule for the n-thterm, and (d) an explicit rule for the n-th term.

96, 48,24, 12,6, 3.

(a) The ratio ( r ) between successive terms is –1/2.

10 110 96 1 2a (b)

3

16

1 96,a (c) 11 2 ,n na a 2n

196 1 2

n

na (d)

Guided PracticeThe second and fifth terms of a sequence are 3 and 24,respectively. Find explicit and recursive formulas for thesequence if it is (a) arithmetic and (b) geometric.

If the sequence is arithmetic:

2 3a a d

5 4 24a a d 3 21d

7d

Explicit Rule:

4 1 7na n

4a

7 11n Recursive Rule:

1 4a 1 7n na a 2n

Guided PracticeThe second and fifth terms of a sequence are 3 and 24,respectively. Find explicit and recursive formulas for thesequence if it is (a) arithmetic and (b) geometric.

If the sequence is geometric:1

2 3a a r 4

5 24a a r 3 1 8r 2r

Explicit Rule:

11.5 2

n

na

1.5a

23 2

n

Recursive Rule:

1 1.5a 12n na a 2n