**
Fibonacci Extended**

*Problem Statement*

Choose two integers. Add them together to create a third integer. Add the
second and third integer of your list to create a fourth. Continue adding the
last two integers to generate a Fibonacci-like sequence, ending with a total
of ten integers. Repeat the process with two different starting integers.

What is the relationship between the seventh term and the sum of the sequence? What is the relationship between the seventh and tenth term of your sequence? Explain.

What is the relationship between the seventh term and the sum of the sequence? What is the relationship between the seventh and tenth term of your sequence? Explain.

*Investigation/Exploration
of the Problem*

Let's choose the integers 1 and 3. If we add these together, we get 4. So our sequence would be the following: 1, 3, 4, 7, 10, 17, ......

Now let's use 2 and 5. If we add these together, we get 7. So our sequence would be: 2, 5, 7, 12, .....

Click here to see the spreadsheet of these sequences.

From the spreadsheet we see that the sum of the sequence is equal to the seventh term multiplied by 11. The same is true when we multiply the seventh term in Fibonacci's Sequence i.e. 13*11 = 143.

Now the tough part....What is the
relationship between the seventh and tenth term of your sequence? This is not
obvious. After collaborating with two classmates, we found that if you multiply
the seventh term by four and then subtract the answer from the tenth term, you
are left with the fourth term. For example, in the first sequence the
**seventh term is 102**, **
the tenth term is 432**** **and the **fourth
term is 24**. Using our "formula" we find
the following:

**102***4
= 408

**432**-408
= **24**

First sequence |

6 |

9 |

15 |

24 |

39 |

63 |

102 |

165 |

267 |

432 |

Click here for the spreadsheet

*Extensions of the Problem*

Would your result be different if you started with negative numbers or fractions?

To find out we simply plug in the the negative number and the fractions in the spreadsheet and we see that the relationship holds true. Click here for the spreadsheet.

*Author &
Contact*

Laura Thomas

Memorial Middle School

Conyers, GA

lthomas@rockdale.k12.ga.us