Question

Consider the Fibonacci sequence 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, .... The first two numbers are 1 and 1. When you add these numbers, you get 2 = 1+1, which becomes the third number in the sequence. When you add the second and third numbers, you get 3 = 1+2, which becomes the fourth number in the sequence. When you add the third and fourth numbers, you get 5 = 2+3, which becomes the fifth number in the sequence; and so on to generate the sequence. Write a short Excel program that generates the first 50 numbers of the Fibonacci sequence, using 1 and 1 as the first two numbers to start the sequence. Put these numbers in Column A in Excel. Also, in the next column in Excel (Column B), divide each number in the sequence by the previous number in the sequence. This column is a new sequence of numbers. Start Column B in Row 2. To what number does the sequence of the quotients in Column B converge? Use the 2-D Bar Chart tool in Excel to graph the first 10 ratios in Column B.

          Consider the Fibonacci sequence 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ....

The first two numbers are 1 and 1. When you add these numbers, you get 2 = 1+1, which becomes the third number in the sequence. When you add the second and third numbers, you get 3 = 1+2, which becomes the fourth number in the sequence. When you add the third and fourth numbers, you get 5 = 2+3, which becomes the fifth number in the sequence; and so on to generate the sequence.

Write a short Excel program that generates the first 50 numbers of the Fibonacci sequence, using 1 and 1 as the first two numbers to start the sequence. Put these numbers in Column A in Excel. Also, in the next column in Excel (Column B), divide each number in the sequence by the previous number in the sequence. This column is a new sequence of numbers. Start Column B in Row 2.

To what number does the sequence of the quotients in Column B converge? Use the 2-D Bar Chart tool in Excel to graph the first 10 ratios in Column B.
        
Show more…

Added by Luis C.

Elementary Statistics a Step by Step Approach
Elementary Statistics a Step by Step Approach
Allan G. Bluman 9th Edition
AceChat toggle button
Close icon
Ace pointing down

Please give Ace some feedback

Your feedback will help us improve your experience

Thumb up icon Thumb down icon
Thanks for your feedback!
Profile picture
Consider the Fibonacci sequence 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, .... The first two numbers are 1 and 1. When you add these numbers, you get 2 = 1+1, which becomes the third number in the sequence. When you add the second and third numbers, you get 3 = 1+2, which becomes the fourth number in the sequence. When you add the third and fourth numbers, you get 5 = 2+3, which becomes the fifth number in the sequence; and so on to generate the sequence. Write a short Excel program that generates the first 50 numbers of the Fibonacci sequence, using 1 and 1 as the first two numbers to start the sequence. Put these numbers in Column A in Excel. Also, in the next column in Excel (Column B), divide each number in the sequence by the previous number in the sequence. This column is a new sequence of numbers. Start Column B in Row 2. To what number does the sequence of the quotients in Column B converge? Use the 2-D Bar Chart tool in Excel to graph the first 10 ratios in Column B.
Close icon
Play audio
Feedback
Powered by NumerAI
Ivan Kochetkov Danielle Fairburn
Kathleen Carty verified

Sri K and 72 other subject Intro Stats / AP Statistics educators are ready to help you.

Ask a new question

*

Labs

-

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

*

Key Concepts

-
Key Concept
Premium Feature
Explore the core concept behind this problem.
Play button
Key Concept
Premium Feature
Explore the core concept behind this problem.
Your browser does not support the video tag.

*

Recommended Videos

-
the-fibonacci-sequence-is-a-well-known-sequence-in-mathematics-that-is-obtained-following-simple-rule-set-the-first-two-entries-of-the-sequence-to-be-0-and-1-and-then-inductively-find-the-ne-91729

The Fibonacci sequence is a well known sequence in mathematics that is obtained following a simple rule: set the first two entries of the sequence to be 0 and 1, and then, inductively, find the next entry by adding two previous entries. This results in the sequence 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, . . . . It is natural to ask whether one can find a formula for the nth Fibonacci number (i.e., the nth entry of this sequence) without having to compute all previous entries. Such rules are called recursions and in this problem we will develop a method to analyze recursion relations. (a) Set F_0 = 0, F_1 = 1, and let F_n denote the nth Fibonacci number (with the convention that 0 is the 0th Fibonacci number). Then the sequence of Fibonacci numbers can be obtained by running the recursion F_n = F_{n-1} + F_{n-2}; F_0 = 0, F_1 = 1. Now, set v_n = [F_n, F_{n-1}]^T. Identify a matrix A such that the recursion above can be written, equivalently, as v_{n+1} = A v_n; v_0 = [F_1, F_0]^T. (b) Using the matrix recursion above A, obtain a (non-recursive) formula for v_{n+1} in terms of powers of A and v_0. (c) Let lambda_1 = (1 + sqrt{5})/2 and lambda_2 = (1 - sqrt{5})/2. Diagonalize A and give a formula for each entry of A^n in terms of lambda_1 and lambda_2. (d) Use your answer from the previous part to obtain a formula for F_n in terms of lambda_1 and lambda_2. (e) Show that lim_{n oinfty} F_{n+1}/F_n = lambda_1. (Note that lambda_1 is the famous "golden ratio").

Adi S.

the-fibonacci-sequence-is-a-famous-sequence-of-numbers-whose-elements-commonly-occur-in-nature-the-t

The Fibonacci sequence is a famous sequence of numbers whose elements commonly occur in nature. The terms in the Fibonacci sequence are $1,1,2,3,5,$ $8,13,21, \ldots$ The ratio of consecutive terms approaches the "golden ratio," $\Phi=\frac{1+\sqrt{5}}{2} .$ If we examine the first digit of the first 85 terms in the Fibonacci sequence, the distribution of digits is as shown in the following table: $$\begin{array}{lccccccccc} \text { Digit } & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ \hline \text { Frequency } & 25 & 16 & 11 & 7 & 7 & 5 & 4 & 6 & 4 \end{array}$$ Is there evidence to support the belief that the first digit of the Fibonacci numbers follow the Benford distribution (shown in Problem 13 ) at the $\alpha=0.05$ level of significance?

Fundamentals of Statistics

Additional Inferential Procedures

Goodness-of-Fit Test

the-fibonacci-sequence-is-a-famous-sequence-of-numbers-whose-elements-commonly-occur-in-nature-the-t

The Fibonacci sequence is a famous sequence of numbers whose elements commonly occur in nature. The terms in the Fibonacci sequence are $1,1,2,3,5,$ $8,13,21, \ldots$ The ratio of consecutive terms approaches the "golden ratio," $\Phi=\frac{1+\sqrt{5}}{2} .$ If we examine the first digit of the first 85 terms in the Fibonacci sequence, the distribution of digits is as shown in the following table: $$\begin{array}{lccccccccc} \text { Digit } & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ \hline \text { Frequency } & 25 & 16 & 11 & 7 & 7 & 5 & 4 & 6 & 4 \end{array}$$ Is there evidence to support the belief that the first digit of the Fibonacci numbers follow the Benford distribution (shown in Problem 13 ) at the $\alpha=0.05$ level of significance?

Fundamentals of Statistics

Additional Inferential Procedures

Goodness-of-Fit Test


*

Recommended Textbooks

-
Elementary Statistics a Step by Step Approach

Elementary Statistics a Step by Step Approach

Allan G. Bluman 9th Edition
achievement 1,163 solutions
The Practice of Statistics for AP

The Practice of Statistics for AP

Daren S. Starnes, Daniel S. Yates, David S. Moore 4th Edition
achievement 1,082 solutions
Introductory Statistics

Introductory Statistics

Barbara Illowsky, Susan Dean 1st Edition
achievement 1,470 solutions

*

Transcript

-
00:02 The first and second row and second row in column are filled with the value i.
00:27 Step 2.
00:29 In the third row, the formula a1 plus a2 is applied...
Need help? Use Ace
Ace is your personal tutor. It breaks down any question with clear steps so you can learn.
Start Using Ace
Ace is your personal tutor for learning
Step-by-step explanations
Instant summaries
Summarize YouTube videos
Understand textbook images or PDFs
Study tools like quizzes and flashcards
Listen to your notes as a podcast
Continue solving this problem
Create a free account to:
  • View full step-by-step solution
  • Ask follow-up questions with Ace AI
  • Save progress and study later
Continue Free
Join the community

18,000,000+

Students on Numerade


Trusted by students at 8,000+ universities

Numerade

Get step-by-step video solution
from top educators

Continue with Clever
or



By creating an account, you agree to the Terms of Service and Privacy Policy
Already have an account? Log In

A free answer
just for you

Watch the video solution with this free unlock.

Numerade

Log in to watch this video
...and 100,000,000 more!


EMAIL

PASSWORD

OR
Continue with Clever