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Questions and Answers
Given the recursive definition of the Fibonacci sequence, $F_1 = 1$, $F_2 = 1$, and $F_n = F_{n-1} + F_{n-2}$ for $n > 2$, what is the value of $F_7$?
Given the recursive definition of the Fibonacci sequence, $F_1 = 1$, $F_2 = 1$, and $F_n = F_{n-1} + F_{n-2}$ for $n > 2$, what is the value of $F_7$?
- 13 (correct)
- 34
- 8
- 21
In the original rabbit problem that led to the discovery of the Fibonacci sequence, which of the following assumptions is NOT made?
In the original rabbit problem that led to the discovery of the Fibonacci sequence, which of the following assumptions is NOT made?
- Females give birth to one male-female pair after mating.
- Rabbits take one month to mature before mating.
- Mating occurs one month after maturation.
- Rabbits can die. (correct)
If $F_n$ represents the nth Fibonacci number, which of the following expressions correctly defines the Fibonacci sequence for $n > 2$?
If $F_n$ represents the nth Fibonacci number, which of the following expressions correctly defines the Fibonacci sequence for $n > 2$?
- $F_n = F_{n-1} - F_{n-2}$
- $F_n = F_{n-1} * F_{n-2}$
- $F_n = F_{n-1} + F_{n-2}$ (correct)
- $F_n = F_{n+1} - F_{n+2}$
A modified Fibonacci sequence starts with the terms 2 and 3. What is the sixth term in this sequence?
A modified Fibonacci sequence starts with the terms 2 and 3. What is the sixth term in this sequence?
Consider a scenario similar to Fibonacci's rabbit problem, but this time, instead of one pair, each pair of rabbits gives birth to two pairs of rabbits (one male-female pair and one female-female pair) after mating. Assuming the initial conditions are the same (one pair of rabbits, one month to mature, etc.), how many pairs of rabbits will there be after 3 months?
Consider a scenario similar to Fibonacci's rabbit problem, but this time, instead of one pair, each pair of rabbits gives birth to two pairs of rabbits (one male-female pair and one female-female pair) after mating. Assuming the initial conditions are the same (one pair of rabbits, one month to mature, etc.), how many pairs of rabbits will there be after 3 months?
Flashcards
Fibonacci Sequence
Fibonacci Sequence
A sequence where each term is the sum of the two preceding terms, starting with 1 and 1.
F₁
F₁
The first Fibonacci number.
F₂
F₂
The second Fibonacci number.
Fₙ (nth Fibonacci number)
Fₙ (nth Fibonacci number)
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Binet's Formula
Binet's Formula
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Study Notes
- The Fibonacci Sequence exhibits a numerical pattern.
- It originated as the answer to the famous rabbit problem.
Leonardo Fibonacci
- Leonardo of Pisa, also known as Fibonacci, is one of the best-known mathematicians of medieval Europe.
- In 1202, after a trip that took him to several Arab and Eastern countries, he wrote the book Liber Abaci.
Rabbit Problem
- Newly born rabbits in field take a month to mature before mating.
- One month after mating, females give birth to one male-female pair and then mate again.
- No rabbits die.
- Determine how many rabbit pairs there are after one year.
- In a sequence of the number of pairs of rabbits after 5 months, you get 1, 1, 2, 3, 5.
- The number of pairs of rabbits of any month after the first two months can be found by adding the numbers of pairs of rabbits in each of the two previous months.
Recursive Definition
- The recursive definition of the Fibonacci sequence is that each number after the first two is the sum of the two preceding ones.
- Let the notation F represent the nth Fibonacci number.
- The numbers in the Fibonacci sequence are given by the following: F₁ = 1, F₂ = 1, and F = F-1 + F-2, where n > 3
Golden Ratio
- The number to which the ratios in the Fibonacci sequence converge is also called the Golden Ratio.
- The Golden Ratio is an irrational number: it has an infinite number of decimal places and it never repeats itself.
- Generally, the Golden Ratio is rounded to 1.618 from (1+√5)/2 = 1.618034...
Binet's Formula
- Binet's Formula allows you to find the nth term in a Fibonacci sequence.
- The formula is: F = (φ^n – (1 – φ)^n) / √5 which relates to the Golden Ratio.
- To find the 9th term of a Fibonacci Sequence using Binet's Formula use the six decimal places of the Golden Ratio (1.618034).
- An example of this is:
- F₉ = (1.618034⁹ – (1 - 1.618034)⁹) / √5
- F₉ = (76.01316037 - (-.618034)⁹) / √5
- F₉ = (76.01316037 - 0.01315561965) / √5
- F₉ = 76.00000475 / √5
- F₉ = 33.98823538
- F₉ = 34
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Description
Explanation of the Fibonacci sequence and its origins with the famous rabbit problem. The Fibonacci sequence follows a pattern where each number is the sum of the two preceding numbers. The sequence starts with 1, 1, 2, 3, 5, and so on.
