Combinatorial Enumeration of Ragas (Scales of Integer Sequences
Lesson 1: Integer Sequences. Student Outcome: You will be able to examine sequences and understand...
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Transcript of Lesson 1: Integer Sequences. Student Outcome: You will be able to examine sequences and understand...
![Page 1: Lesson 1: Integer Sequences. Student Outcome: You will be able to examine sequences and understand the notations used to describe them.](https://reader036.fdocuments.us/reader036/viewer/2022062317/5a4d1b5e7f8b9ab0599ac3b6/html5/thumbnails/1.jpg)
Lesson 1:Integer Sequences
![Page 2: Lesson 1: Integer Sequences. Student Outcome: You will be able to examine sequences and understand the notations used to describe them.](https://reader036.fdocuments.us/reader036/viewer/2022062317/5a4d1b5e7f8b9ab0599ac3b6/html5/thumbnails/2.jpg)
Student Outcome:You will be able to examine sequences and understand the notations used to describe them.
![Page 3: Lesson 1: Integer Sequences. Student Outcome: You will be able to examine sequences and understand the notations used to describe them.](https://reader036.fdocuments.us/reader036/viewer/2022062317/5a4d1b5e7f8b9ab0599ac3b6/html5/thumbnails/3.jpg)
Sequence:an ordered list of elements
Examples:1,3,5,7,9,11,...
.3,.33,.333,.3333,.33333,...
2,4,8,16,32,64
![Page 4: Lesson 1: Integer Sequences. Student Outcome: You will be able to examine sequences and understand the notations used to describe them.](https://reader036.fdocuments.us/reader036/viewer/2022062317/5a4d1b5e7f8b9ab0599ac3b6/html5/thumbnails/4.jpg)
Infinite Sequences-Infinite sequences are sequences where the pattern is regular and continues on infinitely-The notation for an infinite sequence will be denoted by a, “...” at the end of the list
![Page 5: Lesson 1: Integer Sequences. Student Outcome: You will be able to examine sequences and understand the notations used to describe them.](https://reader036.fdocuments.us/reader036/viewer/2022062317/5a4d1b5e7f8b9ab0599ac3b6/html5/thumbnails/5.jpg)
Finite Sequences-finite sequences are sequences where the pattern ends - If it does not have a “...”, then the sequence is considered finite
![Page 6: Lesson 1: Integer Sequences. Student Outcome: You will be able to examine sequences and understand the notations used to describe them.](https://reader036.fdocuments.us/reader036/viewer/2022062317/5a4d1b5e7f8b9ab0599ac3b6/html5/thumbnails/6.jpg)
Infinite or Finite???Classify as finite or infinite.
Ex1) 3,6,9,12,15
Ex2) ½ ,¼,⅛,...
Ex3) 5,7,9,11,13,15
![Page 7: Lesson 1: Integer Sequences. Student Outcome: You will be able to examine sequences and understand the notations used to describe them.](https://reader036.fdocuments.us/reader036/viewer/2022062317/5a4d1b5e7f8b9ab0599ac3b6/html5/thumbnails/7.jpg)
Terms of a Sequence-Each list of numbers is made up of terms.-For instance, given the sequence:
5,10,20,40,80,..
TERM NUMBER SEQUENCE
TERMn TERM
1st Term a1 1 5
2nd Term a2 2 10
3rd Term a3 3 20
4th Term a4 4 40
5th Term a5 5 80
![Page 8: Lesson 1: Integer Sequences. Student Outcome: You will be able to examine sequences and understand the notations used to describe them.](https://reader036.fdocuments.us/reader036/viewer/2022062317/5a4d1b5e7f8b9ab0599ac3b6/html5/thumbnails/8.jpg)
Terms of a SequenceFind the following:Ex1) -3,0,3,6,9,12,... a3= a5=
a7= a10=
![Page 9: Lesson 1: Integer Sequences. Student Outcome: You will be able to examine sequences and understand the notations used to describe them.](https://reader036.fdocuments.us/reader036/viewer/2022062317/5a4d1b5e7f8b9ab0599ac3b6/html5/thumbnails/9.jpg)
Terms of a SequenceFind the following:Ex2) 1,2,4,8,16,... a1= a3=
a6= a9=
![Page 10: Lesson 1: Integer Sequences. Student Outcome: You will be able to examine sequences and understand the notations used to describe them.](https://reader036.fdocuments.us/reader036/viewer/2022062317/5a4d1b5e7f8b9ab0599ac3b6/html5/thumbnails/10.jpg)
nth Term of a Sequence-The nth term of a sequence is the the equation or formula that will produce the sequence.
-Often called the explicit formula, as it relates each term in a sequence directly to its placement in the sequence
- The nth term is denoted by an.
-For purposes of this class n will always begin with 1.
-Be aware that n can begin with 0, such that a0 is the first term in the sequence but again for purposes of this class, the first term will be a1.
![Page 11: Lesson 1: Integer Sequences. Student Outcome: You will be able to examine sequences and understand the notations used to describe them.](https://reader036.fdocuments.us/reader036/viewer/2022062317/5a4d1b5e7f8b9ab0599ac3b6/html5/thumbnails/11.jpg)
nth Term of a Sequence-For instance the nth term (explicit formula) below produces the following sequence.
an=2n+1
3,5,7,9,..
TERM #
SEQUENCE TERM
an=2n+1 SEQUENCE
1 a1 2(1)+1 3
2 a2 2(2)+1 5
3 a3 2(3)+1 7
4 a4 2(4)+1 9
![Page 12: Lesson 1: Integer Sequences. Student Outcome: You will be able to examine sequences and understand the notations used to describe them.](https://reader036.fdocuments.us/reader036/viewer/2022062317/5a4d1b5e7f8b9ab0599ac3b6/html5/thumbnails/12.jpg)
nth Term of a SequenceEx1) Fill in the chart below and give the first five terms of the sequence that is produced from the nth term (explicit formula) below:
an=n2+1
n SEQUENCE TERM
an=n2+1 SEQUENCE
![Page 13: Lesson 1: Integer Sequences. Student Outcome: You will be able to examine sequences and understand the notations used to describe them.](https://reader036.fdocuments.us/reader036/viewer/2022062317/5a4d1b5e7f8b9ab0599ac3b6/html5/thumbnails/13.jpg)
nth Term of a SequenceEx1) Fill in the chart below and give the first five terms of the sequence that is produced from the nth term (explicit formula) below:
an=n2+1
n SEQUENCE TERM
an=n2+1 SEQUENCE
1
2
3
4
5
![Page 14: Lesson 1: Integer Sequences. Student Outcome: You will be able to examine sequences and understand the notations used to describe them.](https://reader036.fdocuments.us/reader036/viewer/2022062317/5a4d1b5e7f8b9ab0599ac3b6/html5/thumbnails/14.jpg)
nth Term of a SequenceEx1) Fill in the chart below and give the first five terms of the sequence that is produced from the nth term (explicit formula) below:
an=n2+1
n SEQUENCE TERM
an=n2+1 SEQUENCE
1 a1
2 a2
3 a3
4 a4
5 a5
![Page 15: Lesson 1: Integer Sequences. Student Outcome: You will be able to examine sequences and understand the notations used to describe them.](https://reader036.fdocuments.us/reader036/viewer/2022062317/5a4d1b5e7f8b9ab0599ac3b6/html5/thumbnails/15.jpg)
nth Term of a SequenceEx1) Fill in the chart below and give the first five terms of the sequence that is produced from the nth term (explicit formula) below:
an=n2+1
n SEQUENCE TERM
an=n2+1 SEQUENCE
1 a1 (1)2+1
2 a2 (2)2+1
3 a3 (3)2+1
4 a4 (4)2+1
5 a5 (5)2+1
![Page 16: Lesson 1: Integer Sequences. Student Outcome: You will be able to examine sequences and understand the notations used to describe them.](https://reader036.fdocuments.us/reader036/viewer/2022062317/5a4d1b5e7f8b9ab0599ac3b6/html5/thumbnails/16.jpg)
nth Term of a SequenceEx1) Fill in the chart below and give the first five terms of the sequence that is produced from the nth term (explicit formula) below:
an=n2+1
n SEQUENCE TERM
an=n2+1 SEQUENCE
1 a1 (1)2+1 2
2 a2 (2)2+1 5
3 a3 (3)2+1 10
4 a4 (4)2+1 17
5 a5 (5)2+1 26
![Page 17: Lesson 1: Integer Sequences. Student Outcome: You will be able to examine sequences and understand the notations used to describe them.](https://reader036.fdocuments.us/reader036/viewer/2022062317/5a4d1b5e7f8b9ab0599ac3b6/html5/thumbnails/17.jpg)
nth Term of a SequenceEx1) Fill in the chart below and give the first five terms of the sequence that is produced from the nth term (explicit formula) below:
an=n2+1
2,5,10,17,26
![Page 18: Lesson 1: Integer Sequences. Student Outcome: You will be able to examine sequences and understand the notations used to describe them.](https://reader036.fdocuments.us/reader036/viewer/2022062317/5a4d1b5e7f8b9ab0599ac3b6/html5/thumbnails/18.jpg)
nth Term of a SequenceEx2) Now find the terms a20 and a30 from the nth term (explicit formula) below:
an=2(n-1)+4
![Page 19: Lesson 1: Integer Sequences. Student Outcome: You will be able to examine sequences and understand the notations used to describe them.](https://reader036.fdocuments.us/reader036/viewer/2022062317/5a4d1b5e7f8b9ab0599ac3b6/html5/thumbnails/19.jpg)
nth Term of a SequenceEx2) Now find the terms a20 and a30 from the nth term (explicit formula) below:
an=2(n-1)+4 -Remember for the term a20 that it is the 20th term in the sequence and n=20.
a20=2(20-1)+4
-Remember for the term a30 that it is the 30th term in the sequence and n=30.
a30=2(30-1)+4
![Page 20: Lesson 1: Integer Sequences. Student Outcome: You will be able to examine sequences and understand the notations used to describe them.](https://reader036.fdocuments.us/reader036/viewer/2022062317/5a4d1b5e7f8b9ab0599ac3b6/html5/thumbnails/20.jpg)
nth Term of a SequenceEx2) Now find the terms a20 and a30 from the nth term (explicit formula) below:
an=2(n-1)+4
a20=52
a30=62
![Page 21: Lesson 1: Integer Sequences. Student Outcome: You will be able to examine sequences and understand the notations used to describe them.](https://reader036.fdocuments.us/reader036/viewer/2022062317/5a4d1b5e7f8b9ab0599ac3b6/html5/thumbnails/21.jpg)
nth Term of a SequenceEx3) Given the sequence below find the nth term (explicit formula) that produces it.(Remember that the first term is a1.)
2,4,6,8,10,12,...
![Page 22: Lesson 1: Integer Sequences. Student Outcome: You will be able to examine sequences and understand the notations used to describe them.](https://reader036.fdocuments.us/reader036/viewer/2022062317/5a4d1b5e7f8b9ab0599ac3b6/html5/thumbnails/22.jpg)
nth Term of a SequenceEx3) Given the sequence below find the nth term (the explicit formula) that produces it.
4,8,12,16,20,...
![Page 23: Lesson 1: Integer Sequences. Student Outcome: You will be able to examine sequences and understand the notations used to describe them.](https://reader036.fdocuments.us/reader036/viewer/2022062317/5a4d1b5e7f8b9ab0599ac3b6/html5/thumbnails/23.jpg)
HOMEWORKSequences WorksheetProblems: 1-20