The numbers in the following sequence are called Fibonacci numbers:
1, 1, 2, 3, 5, 8, 13, 21, 34
If the measures of the sides of a rectangle are consecutive Fibonacci numbers, it is called a Fibonacci rectangle.
a. Find the sum of the areas of the first two Fibonacci rectangles, the first three Fibonacci rectangles, and so on, up to the first nineteen Fibonacci rectangles.
b. Describe a pattern or relationship that exists between the area sums and the Fibonacci numbers.
Alternate Problem 13
The numbers in the sequence 1, 1, 2, 3, 5, 8, 13, 21 , … are called Fibonacci numbers. The sequence is defined so that the starting numbers are 1, 1, and successive numbers are obtained by adding the two previous numbers. The numbers in the sequence 1, 3, 4, 7, 11, 18, 29, 47, 76, … are called Lucas numbers. The starting numbers are 1, 3, and the other numbers are obtained as in the Fibonacci sequence. The terms of the two sequences are given by symbols such as F6 for 8 and L5 for 11.
Examine the values of Ln and Fn for different values of n and find a formula for Ln – Fn in terms of symbols.
Extension Problem 13
The terms in the Fibonacci sequence are defined as follows:
F1 = 1
F2 = 1
Fn + 1 = Fn + Fn – 1
Investigate values of F2n + F2n + 5 for different values of n to find an expression for F298 + F2103.