http://www.ck12.org Chapter 12. Discrete Math
Solution: 0 + 1 + 1 + 2 + 3 + 5 + 8 + 13 + 21 + 34 + 55 = 143
Example B
Write a recursive definition that fits the following sequence.
3 , 7 , 11 , 15 , 18 ,...
Solution:In order to write a recursive definition for a sequence you must define the pattern and state the first
term. With this information, others would be able to replicate your sequence without having seen it for themselves.
a 1 = 3
ak=ak− 1 + 4
Example C
What are the first nine terms of the sequence defined by:
a 1 = 1
ak=^1 k+1?
Solution: 1 , 2 ,^32 ,^53 ,^83 ,^138 ,^2113 ,^3421 ,^5534
Concept Problem Revisited
The Fibonacci sequence is represented by the recursive definition:
a 1 = 0
a 2 = 1
ak=ak− 2 +ak− 1
Vocabulary
Arecursively defined pattern or sequenceis a sequence with terms that are defined based on the prior term(s) in the
sequence.
Anexplicit pattern or sequenceis a sequence with terms that are defined based on the term number.
Guided Practice
- The Lucas sequence is like the Fibonacci sequence except that the starting numbers are 2 and 1 instead of 1 and
- What are the first ten terms of the Lucas sequence?
- Zeckendorf’s Theorem states that every positive integer can be represented uniquely as a sum of nonconsecutive
Fibonacci numbers. What is the Zeckendorf representation of the number 50 and the number 100? - Consider the following pattern generating rule:
If the last number is odd, multiply it by 3 and add 1.
If the last number is even, divide the number by 2.
Repeat.
Try a few different starting numbers and see if you can state what you think always happens.
Answers: - 2, 1 , 3 , 4 , 7 , 11 , 18 , 29 , 47 , 76
- 50= 34 + 13 +3; 100= 89 + 8 + 3