sequences and series
DESCRIPTION
Sequences and Series. It’s all in Section 9.4a!!!. Sequence – an ordered progression of numbers – examples:. Finite Sequence. 1. 2. Infinite Sequences. 3. (unless otherwise specified, the word “sequence” will refer to an infinite sequence). 4. which is sometimes abbreviated. - PowerPoint PPT PresentationTRANSCRIPT
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
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