Concept Byte: The Fibonacci Sequence

For Use With Lesson 9-2

EXTENSION

One famous mathematical sequence is the Fibonacci sequence. You can find each term of the sequence using addition, but the sequence is not arithmetic.

Example

The recursive formula for the Fibonacci sequence is F n = F n − 2 + F n − 1 , f sub n , equals . f sub n minus 2 end sub . plus . f sub n minus 1 end sub . comma  with F 1 = 1 f sub 1 , equals 1  and F 2 = 1 . f sub 2 , equals 1 .  Using the formula, what are the first five terms of the sequence?

F 1 = 1 f sub 1 , equals 1

F 2 = 1 f sub 2 , equals 1

F 3 = F 1 + F 2 = 1 + 1 = 2 f sub 3 , equals . f sub 1 , plus , f sub 2 , equals 1 plus 1 equals 2

F 4 = F 2 + F 3 = 1 + 2 = 3 f sub 4 , equals . f sub 2 , plus , f sub 3 , equals 1 plus 2 equals 3

F 5 = F 3 + F 4 = 2 + 3 = 5 f sub 5 , equals . f sub 3 , plus , f sub 4 , equals 2 plus 3 equals 5

The first five terms of the Fibonacci sequence are 1, 1, 2, 3, 5.

Exercises

1. Nature The numbers of the Fibonacci sequence are often found in other areas, especially nature. Which term of the Fibonacci sequence does each picture represent?
   1. 
   2. 
   3. 
   4. 
2. Find “diagonals” in Pascal's triangle below by starting with the first 1 in each row and moving one row up and one number to the right. For example, the diagonal starting in the fifth row is 1, 3, 1. The diagonal starting in the sixth row is 1, 4, 3. For each diagonal, write the sum of its entries. What pattern do the sums form?

   

3. 1. Generate the first ten terms of the Fibonacci sequence.
   2. Find the sum of the first ten terms of the Fibonacci sequence. Divide the sum by 11. What do you notice?
   3. Open-Ended Choose two numbers other than 1 and 1. Generate a Fibonacci-like sequence from them. Write the first ten terms of your sequence, find the sum, and divide the sum by 11. What do you notice?
   4. Make a Conjecture What is the sum of the first ten terms of any Fibonacci-like sequence?
4. 1. Study the pattern below. Write the next line.
   2. Without calculating, use the pattern to predict the sum of the squares of the first ten terms of the Fibonacci sequence.
   3. Verify the prediction you made in part (b).

   1 2 + 1 2 = 2 = 1 · 2 1 2 + 1 2 + 2 2 = 6 = 2 · 3 1 2 + 1 2 + 2 2 + 3 2 = 15 = 3 · 5 1 2 + 1 2 + 2 2 + 3 2 + 5 2 = 40 = 5 · 8 table with 4 rows and 3 columns , row1 column 1 , 1 squared , plus , 1 squared , equals , column 2 2 equals , column 3 1 middle dot 2 , row2 column 1 , 1 squared , plus , 1 squared , plus , 2 squared , equals , column 2 6 equals , column 3 2 middle dot 3 , row3 column 1 , 1 squared , plus , 1 squared , plus , 2 squared , plus , 3 squared , equals , column 2 15 equals , column 3 3 middle dot 5 , row4 column 1 , 1 squared , plus , 1 squared , plus , 2 squared , plus , 3 squared , plus , 5 squared , equals , column 2 40 equals , column 3 5 middle dot 8 , end table

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