Express Ln as a function of Fibonacci numbers: Hint: Guess the value of Ln (for small n) as a sum...
2. Some facts about Fibonacci sequence: 0,1,1,2,3,5, 8, 13,21,34,55, 89, for n 0 for n 1 F-1 Ffor n22 what is the lest value of n for which F, > 100? what is the least alle urn ir which F > 10002 Let An (F+F2+..Fl/n be the average of the first n Fibonacci numbers. What is the least value of n for which An 10? Find all n for which F, = n, Explain why these are the only cases....
Express the sum as a p-series.
∞n=1(
2^-ln(n)) (Hint: 2−ln(n) = 1/n^ln2)
Identify p
p=?
I would appreciate any help on this problem for discrete math.
Thanks! (:
15. (Q1, P4) Consider the sequence of partial sums of squares of Fibonacci numbers Just to check that we're all on the same page, this sequence starts 1, 2, 6, 15,40, (a) Guess a formula for the nth partial sum, in terms of Fibonacci numbers. (Hint: Write each term as a product.) (b) Prove your formula is correct by mathematical induction. (c) Explain what this problem has...
8. This exercise is a continuation of the previous one. The Lucas numbers Ln are defined by the same relationship as the Fibonacci numbers. Ln+2 = Ln+1 + Ln. However, we begin with Lo = 2 and L-1, which leads to the sequence 2, 1,3,4,7,11,... 「Ln+1 Ln As before, form the vector as a linear combination of vi and v2, eigenvectors of A. Explain why so that a. Xn+1 = Axn. Express X0 b. -(부).. (뷔 Explain why Ln is...
c++ fibonacci code using loops
Here are 8 Fibonacci numbers: 1, 1, 2, 3, 5, 8, 13, 21 Note that the first Fibonacci number is 1, F(1) = 1 The second Fibonacci number is 1, i.e. F(2) = 1 Other Fibonacci numbers in the sequence is the sum of two previous Fibonacci numbers. For example F(3) = F(2) + F(1). In general F(n) = F(n-1) + F(n-2) Write a program to do the following tasks. User entries are shown in...
Discrete Math and Computer Science
I need help with #2 the programming part is in C++ Thank you!
Main topic and problems for the final project The main purpose of the project is to introduce you how to use a computer as a research tool in an Introductory Discrete Mathematics. In this project you will be asked to show how the Fibonacci sequence (F,) is related to Pascal's triangle using the following identities by hand for small n and then...
In mathematics, the Fibonacci numbers are the series of number that exhibit the following pattern: 0,1,1,2,3,5,8,13,21,34,55,89,144,.... In mathematical notation the sequence Fn of Fibonacci number is defined by the following recurrence relation: Fn=Fn-1+Fn-2 With the initial values of F0=0 and F1=1. Thus, the next number in the series is the sum of the previous two numbers. Write a program that asks the user for a positive integer N and generate the Nth Fibonacci number. Your main function should handle user...
Main topic and problems for the final project The main purpose of the project is to introduce you how to use a in an computer as a research tool Introductory Discrete Mathematics. In this project you will be asked to show how the Fibonacci sequence {Fn} is related to Pascal's triangle using the following identities by hand for small and then by computers with large n. Finally, to rove the identity by mathematical arguments, such as inductions or combinatorics. I...
The following
Implementation of the Fibonacci function is a
correct, but inefficient,
def fibonacci(n):
if n <= 2:
return 1
else:
return fib(n - 1) +
fib(n - 2)
In more details, the
code shown runs very slowly for even relatively small values of
n; it can take minutes or hours to compute even the 40th
or 50th Fibonacci number. The code is inefficient because it makes
too many recursive calls. It ends up recomputing each Fibonacci
number many times....
The Lucas numbers L(n) have almost the same definition as the Fibonacci numbers: If n = 1 if n- 2 L(n 1) L(n - 2) if n > 2. 12, as in Theorem 3.6. Prove that L(n)-α, β n for all n E N. Use strong induction Let α = 1 + v/5 and β-- Proof. First, note that and L(2) suppose as inductive hypothesis that L()-α4 β, for all i k, for some k > 2. Then l(k) =...