Download Sequences – Finding and Describing Patterns 1. Draw the 4th

Sequences – Finding and Describing Patterns
1. Draw the 4th Figure for each geometric pattern. Refer to the figures to complete the tables.
Figure #
Area
1
2
st
4
rd
Challenge
nth
5
th
th
9
Perimeter
Figure #
Area
3
nd
8
st
nd
1
rd
2
th
3
Challenge
nth
th
4
5
Perimeter
st
Figure #
# of segments
1
Figure #
Volume
st
nd
rd
2
th
3
th
4
5
Challenge
nth
7
nd
1
rd
2
th
3
th
4
5
Challenge
nth
1
Surface Area
24
You do not need to draw the 4th Figure for this pattern:
Figure #
# of unit squares
1
st
nd
2
13
3
rd
4
th
Challenge
nth
Sequences – Recursive Rules and Explicit Formulas
A recursive rule defines the numbers (terms) in a sequence based on the previous numbers in the sequence. A recursive rule comes with a formula and a seed (or seeds). A seed is the first number of the sequence. It can be written as a 1 (pronounced “a sub one”) where a is the variable representing the number and 1 indicates the position of the number in the sequence.
Example:
recursive rule: AN = AN − 1  2 for N ≥ 2
seed: A1 = 5
sequence: A1 = 5
A2 = A1  2 = 5  2 = 7
A3 = A2  2 = 7  2 = 9
A4 = A3  2 = 9  2 = 11 and so on …
list format: 5, 7, 9, 11, … (notice the 3rd term in the list is A3 = 9 )
An explicit formula is a function equation in which the input is the position number and the output is the term.
Example:
explicit formula: AN = 2 N  3
sequence: A1 = 21  3 = 5
A2 = 2 2   3 = 7
A3 = 2 3  3 = 9
A4 = 2  4  3 = 11 and so on …
list format: 5, 7, 9, 11, … Notice that the sequence 5, 7, 9, 11, … can be expressed both with a recursive rule and an explicit formula. Not all sequences have this property, and some of those that do have very complex explicit formulas. You will learn about one such sequence today.
The advantage of an explicit formula is that if you wanted to find the 100th number in the sequence, you would simply multiply 2 by 100 and add 3 to get 203. If you only had the recursive rule, you would add 2 to the 99th number. What is the 99th number? Well, you add 2 to the 98th number. What is the 98th number? You can see where this is going … 2. Use the given recursive rule to write a list of the first 5 terms in the sequence.
a. recursive rule: AN = 2 A N − 1  1 for N ≥ 2
seed: A1 = 1
A1 = _____
A2 = ________________
A3 = ________________
A4 = ________________
A5 = ________________
_____ , _____ , _____ , _____ , _____ , … b. recursive rule: B N = 3 B N − 1 − 4 for N ≥ 2
seed: B1 = 2
c. Use the same recursive rule as in part (b), but use B1 = 5 as the seed.
B1 = _____
B1 = _____
B2 = ________________
B2 = ________________
B3 = ________________
B3 = ________________
B 4 = ________________
B 4 = ________________
B5 = ________________
B5 = ________________
_____ , _____ , _____ , _____ , _____ , … _____ , _____ , _____ , _____ , _____ , … d. recursive rule: D N = 2 D N − 1 − D N − 2 for N ≥ 3 seeds: D1 = 1 , D2 = 2
D 1 = _____
D 2 = _____
D 3 = ________________
D 4 = ________________
D 5 = ________________
_____ , _____ , _____ , _____ , _____ , … This next sequence is quite famous. Some of you may have seen it before.
e. recursive rule: F N = F N − 1  F N − 2 for N ≥ 3
seeds: F 1 = 1 , F 2 = 1
F 1 = _____
F 2 = _____
F 3 = ________________
F 4 = ________________
F 5 = ________________
F 6 = ________________
_____ , _____ , _____ , _____ , _____ , _____ … 3. Compute each of the following.
a. F 1  F 3  F 5
c. F 10
F5
b. F 1  3  5
d. F F 6
4. Challenge: Each of the sequences in #2 (re­written below) has an explicit formula. See if you can figure them out. Some are easy, some are not, one is quite complicated.
a. recursive rule: AN = 2 A N − 1  1 for N ≥ 2
seed: A1 = 1
_____ , _____ , _____ , _____ , _____ , … explicit formula: AN = _______________________
b. recursive rule: B N = 3 B N − 1 − 4 for N ≥ 2
seed: B1 = 2
_____ , _____ , _____ , _____ , _____ , … explicit formula: B N = _______________________
c. recursive rule: B N = 3 B N − 1 − 4 for N ≥ 2
seed: B1 = 5
_____ , _____ , _____ , _____ , _____ , … explicit formula: B N = _______________________
d. recursive rule: D N = 2 D N − 1 − D N − 2 for N ≥ 3
seeds: D1 = 1 , D2 = 2
_____ , _____ , _____ , _____ , _____ , … explicit formula: D N = _______________________
e. recursive rule: F N = F N − 1  F N − 2 for N ≥ 3
seeds: F 1 = 1 , F 2 = 1
_____ , _____ , _____ , _____ , _____ , _____ … explicit formula: F N = _______________________
5. What is the name of the sequence of numbers in (e)? _________________________________