1104 ch 11 day 4
- 1. 11.1 Sequences & Summation Notation Day Two John 14:15 "If you love me, you will keep my commandments."
- 2. Not all sequences can be defined with an explicit rule.
- 3. Not all sequences can be defined with an explicit rule. A Recursive rule is one where one or more initial terms are given and all subsequent terms are defined using previous terms.
- 4. Not all sequences can be defined with an explicit rule. A Recursive rule is one where one or more initial terms are given and all subsequent terms are defined using previous terms. an
- 5. Not all sequences can be defined with an explicit rule. A Recursive rule is one where one or more initial terms are given and all subsequent terms are defined using previous terms. an next term
- 6. Not all sequences can be defined with an explicit rule. A Recursive rule is one where one or more initial terms are given and all subsequent terms are defined using previous terms. an an+1 next term
- 7. Not all sequences can be defined with an explicit rule. A Recursive rule is one where one or more initial terms are given and all subsequent terms are defined using previous terms. an an+1 previous next term term
- 8. Not all sequences can be defined with an explicit rule. A Recursive rule is one where one or more initial terms are given and all subsequent terms are defined using previous terms. an−1 an an+1 previous next term term
- 9. Not all sequences can be defined with an explicit rule. A Recursive rule is one where one or more initial terms are given and all subsequent terms are defined using previous terms. an−1 an an+1 second previous next term previous term term
- 10. Not all sequences can be defined with an explicit rule. A Recursive rule is one where one or more initial terms are given and all subsequent terms are defined using previous terms. an−2 an−1 an an+1 second previous next term previous term term
- 11. Find the first five terms of the sequence if: a1 = 3 and an = 2an−1 − 1
- 12. Find the first five terms of the sequence if: a1 = 3 and an = 2an−1 − 1 seed
- 13. Find the first five terms of the sequence if: a1 = 3 and an = 2an−1 − 1 seed current = 2(previous)-1
- 14. Find the first five terms of the sequence if: a1 = 3 and an = 2an−1 − 1 seed current = 2(previous)-1 a1 = 3
- 15. Find the first five terms of the sequence if: a1 = 3 and an = 2an−1 − 1 seed current = 2(previous)-1 a1 = 3 a2 = 2 ( 3) − 1 = 5
- 16. Find the first five terms of the sequence if: a1 = 3 and an = 2an−1 − 1 seed current = 2(previous)-1 a1 = 3 a2 = 2 ( 3) − 1 = 5 a3 = 2 ( 5 ) − 1 = 9
- 17. Find the first five terms of the sequence if: a1 = 3 and an = 2an−1 − 1 seed current = 2(previous)-1 a1 = 3 a2 = 2 ( 3) − 1 = 5 a3 = 2 ( 5 ) − 1 = 9 a4 = 2 ( 9 ) − 1 = 17
- 18. Find the first five terms of the sequence if: a1 = 3 and an = 2an−1 − 1 seed current = 2(previous)-1 a1 = 3 a2 = 2 ( 3) − 1 = 5 a3 = 2 ( 5 ) − 1 = 9 a4 = 2 ( 9 ) − 1 = 17 a5 = 2 (17 ) − 1 = 33
- 19. Find the first five terms of the sequence if: a1 = 3 and an = 2an−1 − 1 seed current = 2(previous)-1 a1 = 3 a2 = 2 ( 3) − 1 = 5 a3 = 2 ( 5 ) − 1 = 9 ∴ 3, 5, 9, 17, 33 a4 = 2 ( 9 ) − 1 = 17 a5 = 2 (17 ) − 1 = 33
- 20. The Fibonacci Sequence 1, 1, 2, 3, 5, 8, 13, 21, K is defined recursively this way:
- 21. The Fibonacci Sequence 1, 1, 2, 3, 5, 8, 13, 21, K is defined recursively this way: a1 = 1 a2 = 1 an = an−1 + an−2
- 22. The Fibonacci Sequence 1, 1, 2, 3, 5, 8, 13, 21, K is defined recursively this way: a1 = 1 needs 2 seed values a2 = 1 an = an−1 + an−2
- 23. The Fibonacci Sequence 1, 1, 2, 3, 5, 8, 13, 21, K is defined recursively this way: a1 = 1 needs 2 seed values a2 = 1 an = an−1 + an−2 Changing the seeds changes the numbers in the sequence, but it is still a Fibonacci Sequence.
- 24. Groups: list the first five terms of this sequence: a1 = 4 and an = 2 ( an−1 + 5 )
- 25. Groups: list the first five terms of this sequence: a1 = 4 and an = 2 ( an−1 + 5 ) a1 = 4 a2 = 2 ( 4 + 5 ) = 18 a3 = 2 (18 + 5 ) = 46 ∴ 4, 18, 46, 102, 214 a4 = 2 ( 46 + 5 ) = 102 a5 = 2 (102 + 5 ) = 214
- 26. Together, let’s find a recursive definition for 1, 3, 2, − 1, − 3, − 2, 1, 3, 2, K and you can see this begins to repeat.
- 27. Together, let’s find a recursive definition for 1, 3, 2, − 1, − 3, − 2, 1, 3, 2, K and you can see this begins to repeat. Look for the pattern and how you can use previous terms to generate current terms ... then identify the rule and the seeds. (I bet we can ‘skin the cat’ on this one)
- 28. I’ll write your ideas on the board. 1, 3, 2, − 1, − 3, − 2, 1, 3, 2, K
- 29. I’ll write your ideas on the board. 1, 3, 2, − 1, − 3, − 2, 1, 3, 2, K Here is one solution I have: a1 = 1, a2 = 3 and an = an−1 − an−2
- 30. I’ll write your ideas on the board. 1, 3, 2, − 1, − 3, − 2, 1, 3, 2, K Here is one solution I have: a1 = 1, a2 = 3 and an = an−1 − an−2 Here is another: a1 = 1, a2 = 3, a3 = 2 and an = −an−3
- 31. I’ll write your ideas on the board. 1, 3, 2, − 1, − 3, − 2, 1, 3, 2, K Here is one solution I have: a1 = 1, a2 = 3 and an = an−1 − an−2 Here is another: a1 = 1, a2 = 3, a3 = 2 and an = −an−3 Neither is better than the other. Both work!!
- 32. Groups, find a recursive definition for 21, 17, 13, 9, 5, 1, − 3, K
- 33. Groups, find a recursive definition for 21, 17, 13, 9, 5, 1, − 3, K Here is a correct definition: a1 = 21, and an = an−1 − 4
- 34. HW #2 “Never tell people how to do things. Tell them what to do and they will surprise you with their ingenuity.” George Patton