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Recursive Formulas

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When you look at a pattern, there are many ways to describe it. You can describe patterns explicitly by stating how each term  a_k is obtained from the term number k . You can also describe patterns recursively by stating how each new term  a_k is obtained from the previous term a_{k-1} . Recursion defines an entire sequence based on the first term and the pattern between consecutive terms. The Fibonacci sequence is a famous recursive sequence, but how is it represented using recursion?

0,1,1,3,5,8,13,21,34, \ldots

Watch This

http://www.youtube.com/watch?v=RjsyEWDEQe0  James Sousa: Finding Terms in a Sequence Given the Recursive Formula

Guidance

When most people see a pattern they see how consecutive terms are related to one another. You might describe patterns with phrases like the ones below:

Pattern Recursive Description
3,6,12,24, \ldots “Each term is twice as big as the previous term”
3,6,9,12, \ldots “Each term is three more than the previous term”

Each phrase is a sign of recursive thinking that defines each term as a function of the previous term.

a_k=f(a_{k-1})

In some cases, a recursive formula can be a function of the previous two or three terms. Keep in mind that the downside of a recursively defined sequence is that it is impossible to immediately know the  100^{th} term without knowing the 99^{th} term.

Example A

For the Fibonacci sequence, determine the first eleven terms and the sum of these terms.

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, \ldots

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

a_k=a_{k-1}+4

Example C

What are the first nine terms of the sequence defined by:

a_1=1

a_k =\frac{1}{k}+1?

Solution:  1, 2, \frac{3}{2}, \frac{5}{3}, \frac{8}{3}, \frac{13}{8}, \frac{21}{13}, \frac{34}{21}, \frac{55}{34}

Concept Problem Revisited

The Fibonacci sequence is represented by the recursive definition:

a_1=0

a_2=1

a_k =a_{k-2}+a_{k-1}

Vocabulary

A recursively defined pattern or sequence is a sequence with terms that are defined based on the prior term(s) in the sequence.

An explicit pattern or sequence is a sequence with terms that are defined based on the term number.

Guided Practice

1. The Lucas sequence is like the Fibonacci sequence except that the starting numbers are 2 and 1 instead of 1 and 0. What are the first ten terms of the Lucas sequence?

2. 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?

3. 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:

1. 2,1,3,4,7,11,18,29,47,76

2. 50=34+13+3;\ 100=89+8+3

3. You can choose any starting positive integer you like. Here are the sequences that start with 7 and 15. 

7,22,11,34,17,52,26,13,40,20,10,5,16,8,4,2,1,4,2,1\ldots

15,46,23,70,35,106,53,160,80,40,20,10,5,16,8,4,2,1,4,2,1\ldots

You could make the conjecture that any starting number will eventually lead to the repeating sequence 4, 2, 1.

This problem is called the Collatz Conjecture and is an unproven statement in mathematics. People have used computers to try all the numbers up to  5\times2^{60} and many mathematicians believe it to be true, but since all natural numbers are infinite in number, this test does not constitute a proof.

Practice

Write a recursive definition for each of the following sequences.

1. 3,7,11,15,19,\ldots

2. 3,9,27,81,\ldots

3. 3,6,9,12,15,\ldots

4. 3,6,12,24,48,\ldots

5. 1,4,16,64,\ldots

6. Find the first 6 terms of the following sequence:

b_1=2

b_2=8

b_k =6b_{k-1}-4b_{k-2}

7. Find the first 6 terms of the following sequence:

c_1=4

c_2=18

c_k =2c_{k-1}+5c_{k-2}

Suppose the Fibonacci sequence started with 2 and 5.

8. List the first 10 terms of the new sequence.

9. Find the sum of the first 10 terms of the new sequence.

Write a recursive definition for each of the following sequences. These are trickier!

10. 1,4,13,40,\ldots

11. 1,5,17,53,\ldots

12. 2,11,56,281, \ldots

13. 2,3,6,18,108,\ldots

14 . 4,6,11,18,30,\ldots

15. 7,13,40,106,292,\ldots

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