MATH 380 Fibonacci Numbers Exercise 1

We set the following notation for the course:

Definition 1. The Fibonacci Numbers will be denoted by {u_n} where u₁ = 1, u₂ = 1, and for all n > 2 u_n = u_n-1 + u_n-2. Occasionally we will let u₀ = 0. The same recurrence relation holds. (u₂ = u₁ + u₀)

1. Write in some orderly manner the values of u₁ through u₂₀.

2. Which Fibonacci numbers are even? (Look at the subscripts.) Why does this pattern appear?

3. There are sequences of numbers related to the Fibonacci Numbers. One is the sequence of Lucas Numbers. These are defined by v₁ = 1, v₂ = 3, and for all n > 2 v_n = v_n-1 + v_n-2.

Write in some orderly manner the values of v₁ through v_10. If we were going to define v₀, what should its value be?

4. . In this exercise we are going to try to guess a formula for a certain sum of Fibonacci Numbers. The sum we work with is the sum of the first n Fibonacci numbers with odd subscripts. Find the first 5 such sums and guess what such a sum is equal to in general.

5. Assignment: due Thursday, September 2.

a) Guess a formula for the following two sums:

i)The sum of the first n Fibonacci Numbers with even subscripts:

ii) The sum of the first n Fibonacci Numbers

b) Prove the result found in 4) above.