Chapter 9



Section 9.1: Recursive Sequences

SEQUENCE: Ordered list of numbers

TERMS: Numbers or values in the sequence

• Terms are identified by their location in the sequence (1st, 2nd, 3rd, etc … )

What is the pattern for each of the following sequences?

1) 13, 15, 17, 19, 21, …

Pattern =

2) 4, 12, 36, 108, 324, …

Pattern =

3) 11, 3, -5, -13, -21, …

Pattern =

4) -3, -5, 2, -7, 9, …

Pattern =

RECURSIVE SEQUENCE:

The terms (numbers) in the sequence are based on using ________________________ terms.

o RECURSIVE RULE: _______________________________ statement for pattern of the sequence.

o SEED: _______________ term(s) of the recursive sequence that are ___________ to begin the pattern

General Notation: [pic]

▪ [pic]= Term _________ at a specific location

▪ a = ________________ to name the sequence

▪ n (subscript) = Term _________ in sequence

Rule Notation:

▪ [pic]= value of nth term

▪ [pic]= 1st previous term or ONE Before an

▪ [pic]= 2nd previous term or TWO Before an

Write Recursive Rules for each pattern statement:

1) The next term of the sequence, a, is equal to 3 times the previous term:

2) The pattern, b, is to multiply the previous two terms :

3) The pattern, u, is the previous term plus 9 all over 4:

4) The next value, a, is the 5 less than double the previous term:

For each of the following recursive sequences:

▪ Write a verbal statement for the pattern explaining the recursive rule.

▪ Find the next three terms of the sequence

1) an = an-1 – 3; a1 = 25

Pattern:

2) un = 4un-1; u1 = 3

Pattern:

3) an = 4 (an-1) + 1; a1 = 5

Pattern:

4) bn = 3bn-1 – 11; b1 = 6

Pattern:

5) rn = 7 – 2rn-1; r1 = -1

Pattern:

SPECIAL TYPES OF RECURSIVE SEQUENCES

ARITHMETIC:

• Patterns of ADDITION or SUBTRACTION by a constant value.

• The common difference, d, is value of pattern.

Recursive Rule:

Examples: Find the common difference and recursive rule.

Exp 1: 25, 42, 59, 76 …

Exp 2: 13.7, 11.1, 8.5, 5.9 …

Exp 3: 5.67, 9.05, 12.43, 15.81, …

GEOMETRIC:

• Patterns of MULTIPICATION by a constant value. Reminder all division operations can be written as multiplications with a fraction.

• The common ratio, r, is value of pattern.

Recursive Rule:

Examples: Find the common ratio and recursive rule.

Exp 1: 3, 15, 75, 375, ...

Exp 2: 32, 56, 98, 171.5,…

Exp 3: 72, 36, 18, 9, …

9.1 RECURSIVE SEQUENCES HOMEWORK ASSIGNMENT

Use a separate piece of paper as necessary.

For each sequence: Identify as ARITHMETIC, GEOMETRIC, or NEITHER

i. Find the recursive rule for arithmetic and geometric answers

ii. Find NEXT TWO TERMS of all sequences

1) 20, 30, 45, 67.5, …

2) 2, 9, 5, 12, 8, …

3) 4, -8, 16, - 32, …

4) [pic]

5) -23, -31.5, -40, -48.5, …

6) 1620, 540, 180, 60, …

7) 7, 16, 34, 70, 142, …

8) 3.5, 4.77, 6.04, 7.31, …

9) [pic], [pic], [pic], [pic], …

Find the 4th TERM for each sequence:

10) an = 2(an-1) ; a1 = 15

11) tn = tn-1 – 7; t1 = 11

12) an = 11 - an-1; a1 = -7

13) un = 5un-1 - 6; u1 = 3

14) an = – 3.5an-1; a1 = -8

15) bn = 3bn-1 + 4; b1 = -6

Write the recursive rule for each sequence description:

16) Next term, a, is 7 more than double the previous term.

17) The pattern, r, is the difference of the last term and 4 all over 11.

18) The next value, u, is the product of the last two terms minus 5.

19) The pattern, a, is the sum of the last two terms times 3.

9.1 Fibonacci Sequence:

Fibonacci Sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, …

Let FN represent the Nth term in the Fibonacci sequence. Then…

|F1 |

|N |FN |[pic]= Round 3 decimal places |N |FN |[pic]= Round 3 decimal places |

|2 |F2 = 1 |[pic] |7 |F7 = 13 |[pic] |

|3 |F3 = 2 |[pic] |8 |F8 = 21 |[pic] |

|4 |F4 = 3 |[pic] |9 |F9 = 34 |[pic] |

|5 |F5 = 5 |[pic] |10 |F10 = 55 |[pic] |

|6 |F6 = 8 |[pic] |11 |F11 = 89 |[pic] |

Do you notice any pattern or a limit to the ratios of [pic] as N increases?

Solve the Quadratic Equation: x2 = x + 1

Where have we seen these two solutions?

GOLDEN RATIO (Phi = Ф, φ): [pic]

▪ Also known as divine proportion, golden number, or golden section

POWERS OF THE GOLDEN RATIO: [pic]

Since PHI,[pic], is a solution to the equation x2 = x + 1, obtain the identity Φ2 = Φ + 1.

Φ3 =

Φ4 =

Φ5 =

Φ6 =

OBSERVATIONS ABOUT THE COEFFICIENTS IN OUR FORMULA:

POWERS OF THE GOLDEN RATIO FORMULA: Ф N =

Write all answers in terms of FIBONACCI Numbers or GOLDEN RATIO.

1) [pic]

2) [pic]

3) [pic]

4) Find the value of a and b in each of the following equations:

a. φ10 = aφ + b

b. φ15 = aφ + b

GEOMETRIC SHAPE QUICK FACTS ON FIBONACCI AND GOLDEN RATIO

GOLDEN RECTANGLE. A rectangle whose sides are in the __________________________ of long side length to short side length equals the ________________________ is called a golden rectangle.

FIBONACCI RECTANGLE: If the sides are _____________________________ Fibonacci numbers then it is a Fibonacci rectangle.

Section 9.3 Gnomons

Similarity: Two objects SIMILAR if one is scaled version (proportional in size) of the other.

|TRIANGLE |SQUARE AND CIRCLES |

| | |

| | |

| | |

|RECTANGLE |RINGS |

| | |

| | |

| | |

| | |

| | |

| | |

SIMILARITY and GEOMETRIC SHAPE PROBLEMS:

1) The original rectangle A has length of 5 and width of 8. Rectangle B is 3 times larger than the rectangle A. Rectangle C is half the size of the rectangle A.

| |Rectangle A |Rectangle B |Rectangle C |

|Dimensions |5 by 8 | | |

|Perimeter | | | |

|Area | | | |

2) If R and R’ are similar rectangles, and the length of R’ is 2.5 times larger than R.

a. If the perimeter of R is 23 in, what is the perimeter of R’?

b.

c. If the perimeter of R’ is 23 in,

what is the perimeter of R?

d. If the area of R is 45 in2, what is the area of R’?

e. If the area of R’ is 45 in2, what is the area of R?

3) R is a rectangle with length = x and width = 4. R’ is a rectangle with length = 10 + x and width = 9. Determine the value of x that makes R and R’ similar.

4) Two triangles are known to be similar. One triangle has sides of 3, 4, 5. It is only known that the middle side length of the similar triangle is 14.4. What are the other side lengths?

5) O and O’ are similar rings. The inner radius of O is 6 and the inner radius of O’ 12 is and the outer radius of O’ is 24. What is the area of ring O?

6) An isosceles triangle has side lengths of 14 and 18. A second isosceles triangle has side lengths of 63 and 77. Are these two triangles similar?

GNOMONS:

Given an original figure A, then a gnomon G to figure A is a figure that when suitably attached to A produces a new figure G&A. The new figure G&A is SIMILAR to A.

▪ & = suitably attached

Gnomons of Different Shapes: A is the original object and G is the Gnomon being attached, and G&A is new similar object to A.

OBSERVATIONS:

DRAW YOUR OWN GNOMONS: Sketch a gnomon to create a larger similar shape.

CLASS WORK PROBLEMS:

1) Find the value of x so that the shaded rectangle is a gnomon to the white rectangle. (Not Drawn to scale)

i. What is the area of the shaded region?

[pic]

2) Find the value of x so that the shaded region is a gnomon to the white square that is 7 by 7. (Not Drawn to scale)

i. What is the area of the shaded region?

[pic]

3) Find the value of x and y so that the shaded triangle is a gnomon to the white triangle. (Not Drawn to scale)

[pic]

4) Find the value of x and y so that the shaded region is a gnomon to the white right triangle. (Not Drawn to scale)

i. What is the area of the shaded region?

[pic]

HOMEWORK: pp. 332 – 334 #36 – 48 (EVEN), 49, 50 (Optional: 60 – 63)[pic]

-----------------------

B1

C1

A1

c2

b2

a2

r

R

S1

s2

W1

w2

L1

l2

Ri

RO

rO

ri

ENLARGE

(scale factor = k)

LENGTHS:

AREAS:

VOLUME:

SHRINK

(scale factor = k)

LENGTHS:

AREAS:

VOLUME:

SQUARE

TRIANGLE

CIRCLE

A

A

G

G

G

A

RECTANGLE:

RECTANGLE:

A

G

A

G

9

36

x

6

5

x

7.5

12

9

6

x

y

5

12

13

5

x

y

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