Download presentation

Presentation is loading. Please wait.

Published byClare Barber Modified over 7 years ago

1
11.5 = Recursion & Iteration

2
Arithmetic = adding (positive or negative)

3
3, 6, 9, 12, …

4
Arithmetic = adding (positive or negative) 3, 6, 9, 12, … d = 3

5
Arithmetic = adding (positive or negative) 3, 6, 9, 12, … d = 3 *Formula for the n th term based on a 1 and d. a n = a 1 +(n–1)d

6
Arithmetic = adding (positive or negative) 3, 6, 9, 12, … d = 3 *Formula for the n th term based on a 1 and d. a n = a 1 +(n–1)d Geometric = multiplying (#’s > 1 or #’s < 1)

7
Arithmetic = adding (positive or negative) 3, 6, 9, 12, … d = 3 *Formula for the n th term based on a 1 and d. a n = a 1 +(n–1)d Geometric = multiplying (#’s > 1 or #’s < 1) 2, 10, 50, 250, …

8
Arithmetic = adding (positive or negative) 3, 6, 9, 12, … d = 3 *Formula for the n th term based on a 1 and d. a n = a 1 +(n–1)d Geometric = multiplying (#’s > 1 or #’s < 1) 2, 10, 50, 250, … r = 5

9
Arithmetic = adding (positive or negative) 3, 6, 9, 12, … d = 3 *Formula for the n th term based on a 1 and d. a n = a 1 +(n–1)d Geometric = multiplying (#’s > 1 or #’s < 1) 2, 10, 50, 250, … r = 5 *Formula for the n th term based on a 1 and r. a n = a 1 r (n – 1)

10
Arithmetic = adding (positive or negative) 3, 6, 9, 12, … d = 3 *Formula for the n th term based on a 1 and d. a n = a 1 +(n–1)d Geometric = multiplying (#’s > 1 or #’s < 1) 2, 10, 50, 250, … r = 5 *Formula for the n th term based on a 1 and r. a n = a 1 r (n – 1) Recursion = formula-based (“neither”)

11
Arithmetic = adding (positive or negative) 3, 6, 9, 12, … d = 3 *Formula for the n th term based on a 1 and d. a n = a 1 +(n–1)d Geometric = multiplying (#’s > 1 or #’s < 1) 2, 10, 50, 250, … r = 5 *Formula for the n th term based on a 1 and r. a n = a 1 r (n – 1) Recursion = formula-based (“neither”) 2, 4, 16, 256, …

12
Arithmetic = adding (positive or negative) 3, 6, 9, 12, … d = 3 *Formula for the n th term based on a 1 and d. a n = a 1 +(n–1)d Geometric = multiplying (#’s > 1 or #’s < 1) 2, 10, 50, 250, … r = 5 *Formula for the n th term based on a 1 and r. a n = a 1 r (n – 1) Recursion = formula-based (“neither”) 2, 4, 16, 256, … -The pattern is that you’re squaring each previous term.

13
Arithmetic = adding (positive or negative) 3, 6, 9, 12, … d = 3 *Formula for the n th term based on a 1 and d. a n = a 1 +(n–1)d Geometric = multiplying (#’s > 1 or #’s < 1) 2, 10, 50, 250, … r = 5 *Formula for the n th term based on a 1 and r. a n = a 1 r (n – 1) Recursion = formula-based (“neither”) 2, 4, 16, 256, … -The pattern is that you’re squaring each previous term. a n+1 = (a n ) 2

14
Recursion = formula-based (“neither”) 2, 4, 16, 256, … -The pattern is that you’re squaring each previous term. a n+1 = (a n ) 2 *Note that this formula only applies to this particular example!!!

15
Recursion = formula-based (“neither”) 2, 4, 16, 256, … -The pattern is that you’re squaring each previous term. a n+1 = (a n ) 2 *Note that this formula only applies to this particular example!!! Basic Formula: next term = #(1 st term) # ± #

16
Recursion = formula-based (“neither”) 2, 4, 16, 256, … -The pattern is that you’re squaring each previous term. a n+1 = (a n ) 2 *Note that this formula only applies to this particular example!!! Basic Formula: next term = #(1 st term) # ± # **The #’s are possibilities, but not requirements.

17
Recursion = formula-based (“neither”) 2, 4, 16, 256, … -The pattern is that you’re squaring each previous term. a n+1 = (a n ) 2 *Note that this formula only applies to this particular example!!! Basic Formula: next term = #(1 st term) # ± # **The #’s are possibilities, but not requirements. Exs. a n = 3a n-1 + 4 a n+1 = (a n ) 2 – 9 a n+2 = 2a n – a n+1

18
Ex. 1 Find the first five terms of each sequence. a 1 = 10, a n+1 = 4a n + 1

19
Ex. 1 Find the first five terms of each sequence. a 1 = 10, a n+1 = 4a n + 1 a 1 = 10

20
Ex. 1 Find the first five terms of each sequence. a 1 = 10, a n+1 = 4a n + 1 a 1 = 10 a 1+1 = 4a 1 + 1

21
Ex. 1 Find the first five terms of each sequence. a 1 = 10, a n+1 = 4a n + 1 a 1 = 10 a 1+1 = 4a 1 + 1 = 4(10) + 1

22
Ex. 1 Find the first five terms of each sequence. a 1 = 10, a n+1 = 4a n + 1 a 1 = 10 a 1+1 = 4a 1 + 1 = 4(10) + 1 = 41

23
Ex. 1 Find the first five terms of each sequence. a 1 = 10, a n+1 = 4a n + 1 a 1 = 10 a 1+1 = 4a 1 + 1 = 4(10) + 1 = 41 a 2 = 41

24
Ex. 1 Find the first five terms of each sequence. a 1 = 10, a n+1 = 4a n + 1 a 1 = 10 a 1+1 = 4a 1 + 1 = 4(10) + 1 = 41 a 2 = 41 a 2+1 = 4a 2 + 1

25
Ex. 1 Find the first five terms of each sequence. a 1 = 10, a n+1 = 4a n + 1 a 1 = 10 a 1+1 = 4a 1 + 1 = 4(10) + 1 = 41 a 2 = 41 a 2+1 = 4a 2 + 1 = 4(41) + 1

26
Ex. 1 Find the first five terms of each sequence. a 1 = 10, a n+1 = 4a n + 1 a 1 = 10 a 1+1 = 4a 1 + 1 = 4(10) + 1 = 41 a 2 = 41 a 2+1 = 4a 2 + 1 = 4(41) + 1 = 165

27
Ex. 1 Find the first five terms of each sequence. a 1 = 10, a n+1 = 4a n + 1 a 1 = 10 a 1+1 = 4a 1 + 1 = 4(10) + 1 = 41 a 2 = 41 a 2+1 = 4a 2 + 1 = 4(41) + 1 = 165 a 3 = 165

28
Ex. 1 Find the first five terms of each sequence. a 1 = 10, a n+1 = 4a n + 1 a 1 = 10 a 1+1 = 4a 1 + 1 = 4(10) + 1 = 41 a 2 = 41 a 2+1 = 4a 2 + 1 = 4(41) + 1 = 165 a 3 = 165 a 3+1 = 4a 3 + 1

29
Ex. 1 Find the first five terms of each sequence. a 1 = 10, a n+1 = 4a n + 1 a 1 = 10 a 1+1 = 4a 1 + 1 = 4(10) + 1 = 41 a 2 = 41 a 2+1 = 4a 2 + 1 = 4(41) + 1 = 165 a 3 = 165 a 3+1 = 4a 3 + 1 = 4(165) + 1

30
Ex. 1 Find the first five terms of each sequence. a 1 = 10, a n+1 = 4a n + 1 a 1 = 10 a 1+1 = 4a 1 + 1 = 4(10) + 1 = 41 a 2 = 41 a 2+1 = 4a 2 + 1 = 4(41) + 1 = 165 a 3 = 165 a 3+1 = 4a 3 + 1 = 4(165) + 1 = 661

31
Ex. 1 Find the first five terms of each sequence. a 1 = 10, a n+1 = 4a n + 1 a 1 = 10 a 1+1 = 4a 1 + 1 = 4(10) + 1 = 41 a 2 = 41 a 2+1 = 4a 2 + 1 = 4(41) + 1 = 165 a 3 = 165 a 3+1 = 4a 3 + 1 = 4(165) + 1 = 661 a 4 = 661

32
Ex. 1 Find the first five terms of each sequence. a 1 = 10, a n+1 = 4a n + 1 a 1 = 10 a 1+1 = 4a 1 + 1 = 4(10) + 1 = 41 a 2 = 41 a 2+1 = 4a 2 + 1 = 4(41) + 1 = 165 a 3 = 165 a 3+1 = 4a 3 + 1 = 4(165) + 1 = 661 a 4 = 661 a 4+1 = 4a 4 + 1

33
Ex. 1 Find the first five terms of each sequence. a 1 = 10, a n+1 = 4a n + 1 a 1 = 10 a 1+1 = 4a 1 + 1 = 4(10) + 1 = 41 a 2 = 41 a 2+1 = 4a 2 + 1 = 4(41) + 1 = 165 a 3 = 165 a 3+1 = 4a 3 + 1 = 4(165) + 1 = 661 a 4 = 661 a 4+1 = 4a 4 + 1 = 4(661) + 1

34
Ex. 1 Find the first five terms of each sequence. a 1 = 10, a n+1 = 4a n + 1 a 1 = 10 a 1+1 = 4a 1 + 1 = 4(10) + 1 = 41 a 2 = 41 a 2+1 = 4a 2 + 1 = 4(41) + 1 = 165 a 3 = 165 a 3+1 = 4a 3 + 1 = 4(165) + 1 = 661 a 4 = 661 a 4+1 = 4a 4 + 1 = 4(661) + 1 = 2645

35
Ex. 1 Find the first five terms of each sequence. a 1 = 10, a n+1 = 4a n + 1 a 1 = 10 a 1+1 = 4a 1 + 1 = 4(10) + 1 = 41 a 2 = 41 a 2+1 = 4a 2 + 1 = 4(41) + 1 = 165 a 3 = 165 a 3+1 = 4a 3 + 1 = 4(165) + 1 = 661 a 4 = 661 a 4+1 = 4a 4 + 1 = 4(661) + 1 = 2645 a 5 = 2645

36
Ex. 1 Find the first five terms of each sequence. a 1 = 10, a n+1 = 4a n + 1 a 1 = 10 a 1+1 = 4a 1 + 1 = 4(10) + 1 = 41 a 2 = 41 a 2+1 = 4a 2 + 1 = 4(41) + 1 = 165 a 3 = 165 a 3+1 = 4a 3 + 1 = 4(165) + 1 = 661 a 4 = 661 a 4+1 = 4a 4 + 1 = 4(661) + 1 = 2645 a 5 = 2645

37
Ex. 2 Write a recursive formula for the sequence. 16, 10, 7, 5.5, 4.75

38
Ex. 2 Write a recursive formula for the sequence. 16 10 7 5.5 4.75 -6 -3 -1.5 -0.75

39
Ex. 2 Write a recursive formula for the sequence. 16 10 7 5.5 4.75 -6 -3 -1.5 -0.75 *Each difference is half the previous difference!

40
Ex. 2 Write a recursive formula for the sequence. 16 10 7 5.5 4.75 -6 -3 -1.5 -0.75 *Each difference is half the previous difference! a 1 = 16

41
Ex. 2 Write a recursive formula for the sequence. 16 10 7 5.5 4.75 -6 -3 -1.5 -0.75 *Each difference is half the previous difference! a 1 = 16 a 2 = 0.5(16) ± ? = 10

42
Ex. 2 Write a recursive formula for the sequence. 16 10 7 5.5 4.75 -6 -3 -1.5 -0.75 *Each difference is half the previous difference! a 1 = 16 a 2 = 0.5(16) ± ? = 10 8 ± ? = 10

43
Ex. 2 Write a recursive formula for the sequence. 16 10 7 5.5 4.75 -6 -3 -1.5 -0.75 *Each difference is half the previous difference! a 1 = 16 a 2 = 0.5(16) ± ? = 10 8 ± ? = 10 a 3 = 0.5(10) ± ? = 7

44
Ex. 2 Write a recursive formula for the sequence. 16 10 7 5.5 4.75 -6 -3 -1.5 -0.75 *Each difference is half the previous difference! a 1 = 16 a 2 = 0.5(16) ± ? = 10 8 ± ? = 10 a 3 = 0.5(10) ± ? = 7 5 ± ? = 7

45
Ex. 2 Write a recursive formula for the sequence. 16 10 7 5.5 4.75 -6 -3 -1.5 -0.75 *Each difference is half the previous difference! a 1 = 16 a 2 = 0.5(16) ± ? = 10 8 ± ? = 10 a 3 = 0.5(10) ± ? = 7 5 ± ? = 7 a 4 = 0.5(7) ± ? = 5.5

46
Ex. 2 Write a recursive formula for the sequence. 16 10 7 5.5 4.75 -6 -3 -1.5 -0.75 *Each difference is half the previous difference! a 1 = 16 a 2 = 0.5(16) ± ? = 10 8 ± ? = 10 a 3 = 0.5(10) ± ? = 7 5 ± ? = 7 a 4 = 0.5(7) ± ? = 5.5 3.5 ± ? = 5.5

47
Ex. 2 Write a recursive formula for the sequence. 16 10 7 5.5 4.75 -6 -3 -1.5 -0.75 *Each difference is half the previous difference! a 1 = 16 a 2 = 0.5(16) ± ? = 10 8 ± ? = 10 a 3 = 0.5(10) ± ? = 7 5 ± ? = 7 a 4 = 0.5(7) ± ? = 5.5 3.5 ± ? = 5.5 a 5 = 0.5(5.5) ± ? = 4.75

48
Ex. 2 Write a recursive formula for the sequence. 16 10 7 5.5 4.75 -6 -3 -1.5 -0.75 *Each difference is half the previous difference! a 1 = 16 a 2 = 0.5(16) ± ? = 10 8 ± ? = 10 a 3 = 0.5(10) ± ? = 7 5 ± ? = 7 a 4 = 0.5(7) ± ? = 5.5 3.5 ± ? = 5.5 a 5 = 0.5(5.5) ± ? = 4.75 2.75 ± ? = 4.75

49
Ex. 2 Write a recursive formula for the sequence. 16 10 7 5.5 4.75 -6 -3 -1.5 -0.75 *Each difference is half the previous difference! a 1 = 16 a 2 = 0.5(16) ± ? = 10 8 ± 2 = 10 a 3 = 0.5(10) ± ? = 7 5 ± 2 = 7 a 4 = 0.5(7) ± ? = 5.5 3.5 ± 2 = 5.5 a 5 = 0.5(5.5) ± ? = 4.75 2.75 ± 2 = 4.75

50
Ex. 2 Write a recursive formula for the sequence. 16 10 7 5.5 4.75 -6 -3 -1.5 -0.75 *Each difference is half the previous difference! a 1 = 16 a 2 = 0.5(16) ± ? = 10 8 ± 2 = 10 a 3 = 0.5(10) ± ? = 7 5 ± 2 = 7 a 4 = 0.5(7) ± ? = 5.5 3.5 ± 2 = 5.5 a 5 = 0.5(5.5) ± ? = 4.75 2.75 ± 2 = 4.75So a n+1 = 0.5a n + 2

51
Ex. 2 Write a recursive formula for the sequence. 16 10 7 5.5 4.75 -6 -3 -1.5 -0.75 *Each difference is half the previous difference! a 1 = 16 a 2 = 0.5(16) ± ? = 10 8 ± 2 = 10 a 3 = 0.5(10) ± ? = 7 5 ± 2 = 7 a 4 = 0.5(7) ± ? = 5.5 3.5 ± 2 = 5.5 a 5 = 0.5(5.5) ± ? = 4.75 2.75 ± 2 = 4.75So a n+1 = 0.5a n + 2

52
Iteration = proceeding terms of a recursive sequence

53
Iteration = proceeding terms of a recursive sequence Ex. 3 Find the first three iterates of the function for the given initial value. f(x) = -6x + 3x 0 = 8

54
Iteration = proceeding terms of a recursive sequence Ex. 3 Find the first three iterates of the function for the given initial value. f(x) = -6x + 3x 0 = 8 a 1 = -6(8) + 3

55
Iteration = proceeding terms of a recursive sequence Ex. 3 Find the first three iterates of the function for the given initial value. f(x) = -6x + 3x 0 = 8 a 1 = -6(8) + 3 = -45

56
Iteration = proceeding terms of a recursive sequence Ex. 3 Find the first three iterates of the function for the given initial value. f(x) = -6x + 3x 0 = 8 a 1 = -6(8) + 3 = -45 a 2 = -6(-45) + 3

57
Iteration = proceeding terms of a recursive sequence Ex. 3 Find the first three iterates of the function for the given initial value. f(x) = -6x + 3x 0 = 8 a 1 = -6(8) + 3 = -45 a 2 = -6(-45) + 3 = 273

58
Iteration = proceeding terms of a recursive sequence Ex. 3 Find the first three iterates of the function for the given initial value. f(x) = -6x + 3x 0 = 8 a 1 = -6(8) + 3 = -45 a 2 = -6(-45) + 3 = 273 a 3 = -6(273) + 3

59
Iteration = proceeding terms of a recursive sequence Ex. 3 Find the first three iterates of the function for the given initial value. f(x) = -6x + 3x 0 = 8 a 1 = -6(8) + 3 = -45 a 2 = -6(-45) + 3 = 273 a 3 = -6(273) + 3 = -1635

60
Iteration = proceeding terms of a recursive sequence Ex. 3 Find the first three iterates of the function for the given initial value. f(x) = -6x + 3x 0 = 8 a 1 = -6(8) + 3 = -45 a 2 = -6(-45) + 3 = 273 a 3 = -6(273) + 3 = -1635

Similar presentations

© 2022 SlidePlayer.com Inc.

All rights reserved.

To make this website work, we log user data and share it with processors. To use this website, you must agree to our Privacy Policy, including cookie policy.

Ads by Google