Fibonacci's Rabbits: Counting Pairs
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Questions and Answers

What is the notation for the nth element of a sequence?

  • a(n)
  • a^n
  • anˆ
  • an (correct)

    • What is the notation for the sequence?

  • {an }∞ n=1 (correct)
  • an from n=1 to ∞
  • {an }
  • an from 1 to ∞

    • What is the limit of the sequence {an }∞ n=1 = {0.1, 0.01, 0.001, 0.0001, · · · }?

  • 10
  • 1
  • 0 (correct)
  • 0.1

    • What does the limit of a sequence represent?

    <p>The number that the sequence approaches as much as we like (A)</p> Signup and view all the answers

    What is the notation for the limit of a sequence?

    <p>lim an = L (A)</p> Signup and view all the answers

    What is true about the sequence {cn }∞ n=1 = {1, 2, 3, 4, · · · }?

    <p>It has no limit (C)</p> Signup and view all the answers

    What is the property of limits that involves a constant number k?

    <p>The limit of a constant multiple is the constant multiple of the limit (A)</p> Signup and view all the answers

    What is the purpose of the notation n→∞?

    <p>To indicate that the sequence approaches a limit as n gets arbitrarily large (C)</p> Signup and view all the answers

    What is the name of the sequence that the number of rabbits evolves as?

    <p>Fibonacci Sequence (B)</p> Signup and view all the answers

    What is the pattern of the number of rabbits in each month?

    <p>1, 1, 2, 3, 5, ... (B)</p> Signup and view all the answers

    What is a collection of numbers called that is 'countably infinite'?

    <p>Sequence (B)</p> Signup and view all the answers

    In which part of the text is the definition of 'sequence' found?

    <p>Part II: Sequences (B)</p> Signup and view all the answers

    What is the name of the author of the text 'Fibonacci's Rabbits'?

    <p>Deniz Karlı (B)</p> Signup and view all the answers

    What natural pattern is an example of the Fibonacci Sequence?

    <p>All of the above (D)</p> Signup and view all the answers

    What is the name of the university mentioned in the text?

    <p>IŞIK UNIVERSITY (C)</p> Signup and view all the answers

    What is the pattern of the number of rabbit pairs in each month?

    <p>1, 1, 2, 3, 5, ... (C)</p> Signup and view all the answers

    What is the ratio of each term to the previous term in the Fibonacci sequence?

    <p>a variable ratio (B)</p> Signup and view all the answers

    What is the relationship between an+2 and an+1 in the Fibonacci sequence?

    <p>an+2 is the sum of an+1 and an (C)</p> Signup and view all the answers

    What is the limit of the ratio sequence, assuming it exists?

    <p>The golden ratio (D)</p> Signup and view all the answers

    What is the relationship between bn+1 and bn?

    <p>bn+1 = 1 + bn (D)</p> Signup and view all the answers

    What is the value of an+1 in terms of an and bn?

    <p>an+1 = an / (bn - 1) (C)</p> Signup and view all the answers

    Why is the ratio sequence useful in the study of the Fibonacci sequence?

    <p>It helps us understand the asymptotic behavior of the Fibonacci sequence (C)</p> Signup and view all the answers

    What is the assumption made in the derivation of the limit of the ratio sequence?

    <p>The ratio sequence has a limit (A)</p> Signup and view all the answers

    What is the value of L in the equation L2 − L − 1 = 0?

    <p>1.61803398875 (B)</p> Signup and view all the answers

    What is the recursive formula for the Fibonacci sequence?

    <p>an+2 = an+1 + an (D)</p> Signup and view all the answers

    What is the formula for the solutions of a quadratic equation ax2 + bx + c = 0?

    <p>±(b + √(b2 - 4ac)) / 2a (D)</p> Signup and view all the answers

    What is the name of the special ratio studied in the text?

    <p>Golden Ratio (C)</p> Signup and view all the answers

    What is the limit of the ratio sequence bn as n approaches infinity?

    <p>φ (A)</p> Signup and view all the answers

    What is the value of the discriminant (∆) of the quadratic equation L2 - L - 1 = 0?

    <p>5 (D)</p> Signup and view all the answers

    What is the application of the Golden Ratio mentioned in the text?

    <p>Golden Rectangle (A)</p> Signup and view all the answers

    What is the rule of the classical Fibonacci sequence?

    <p>an+2 = an+1 + an (D)</p> Signup and view all the answers

    What is the sequence of numbers used to draw the Golden Spiral?

    <p>1, 1, 2, 3, 5, 8, 13, 21,... (C)</p> Signup and view all the answers

    What is the purpose of using the Fibonacci sequence in drawing the Golden Spiral?

    <p>To create a mathematically-derived aesthetic design (C)</p> Signup and view all the answers

    In what direction are the squares drawn to obtain the Golden Spiral?

    <p>Clockwise direction (C)</p> Signup and view all the answers

    Which of the following is a characteristic of the Golden Spiral?

    <p>It gets wider by a factor of the golden ratio for each quarter turn (B)</p> Signup and view all the answers

    What is the length of the side of the first square in the Golden Spiral?

    <p>1 (A)</p> Signup and view all the answers

    What is the purpose of recalling the terms of the Fibonacci sequence?

    <p>To draw the Golden Spiral (C)</p> Signup and view all the answers

    What is the relationship between the side lengths of consecutive squares in the Golden Spiral?

    <p>They are consecutive terms in the Fibonacci sequence (B)</p> Signup and view all the answers

    What is the resulting pattern obtained by drawing squares of side lengths equal to terms of the Fibonacci sequence in a clockwise direction?

    <p>A Golden Spiral pattern (C)</p> Signup and view all the answers

    What is the ratio of the sides of the Golden Rectangle?

    <p>φ (A)</p> Signup and view all the answers

    What is the shape of older TVs that were produced?

    <p>Rectangle with a 4:3 ratio (D)</p> Signup and view all the answers

    What is the purpose of drawing squares of side lengths equal to terms of the Fibonacci sequence?

    <p>To create a Golden Spiral (A)</p> Signup and view all the answers

    What is the approximate value of the Golden Ratio?

    <p>1.77 (A)</p> Signup and view all the answers

    What is the sequence used to draw the Golden Spiral?

    <p>Fibonacci sequence (D)</p> Signup and view all the answers

    What is the direction in which the squares are drawn to obtain the Golden Spiral?

    <p>Clockwise (B)</p> Signup and view all the answers

    What is the characteristic of the Golden Rectangle that makes it pleasing to the human eye?

    <p>It has a ratio of φ (B)</p> Signup and view all the answers

    What is the ratio of newer TVs that were produced?

    <p>16:9 (C)</p> Signup and view all the answers

    What is the relationship between an and an+1 in the Fibonacci sequence?

    <p>an+1 = an + an (A)</p> Signup and view all the answers

    What is the value of bn in terms of an and an+1?

    <p>bn = an+1/an (B)</p> Signup and view all the answers

    What is the assumption made about the limit of the ratio sequence bn?

    <p>It is assumed to exist (D)</p> Signup and view all the answers

    What is the relationship between bn+1 and bn?

    <p>bn+1 = 1 + bn (A)</p> Signup and view all the answers

    What is the pattern of the Fibonacci sequence?

    <p>Each term is the sum of the two previous terms (B)</p> Signup and view all the answers

    What is the purpose of using the ratio sequence bn?

    <p>To study the properties of the Fibonacci sequence (D)</p> Signup and view all the answers

    What is the value of an+2 in terms of an and an+1?

    <p>an+2 = an+1 + an (C)</p> Signup and view all the answers

    What is the significance of the limit of the ratio sequence bn?

    <p>It represents the ratio of the number of rabbit pairs in the nth month to the number of rabbit pairs in the (n-1)th month (B)</p> Signup and view all the answers

    What is the purpose of connecting opposite corners of the squares with arcs in the Golden Spiral?

    <p>To form a connected path (B)</p> Signup and view all the answers

    What is the first step in obtaining the Golden Spiral?

    <p>Drawing squares of side lengths equal to terms of the Fibonacci sequence (B)</p> Signup and view all the answers

    What is the pattern observed in the Golden Spiral?

    <p>A connected path formed by arcs (A)</p> Signup and view all the answers

    Why is the Fibonacci sequence used in drawing the Golden Spiral?

    <p>To create a aesthetically pleasing pattern (D)</p> Signup and view all the answers

    What is the characteristic of the squares in the Golden Spiral?

    <p>They have side lengths equal to terms of the Fibonacci sequence (C)</p> Signup and view all the answers

    What is the result of drawing squares of side lengths equal to terms of the Fibonacci sequence?

    <p>A connected path formed by arcs (C)</p> Signup and view all the answers

    What is the purpose of recalling the terms of the Fibonacci sequence?

    <p>To draw squares of side lengths equal to terms of the Fibonacci sequence (B)</p> Signup and view all the answers

    What is the direction in which the squares are drawn to obtain the Golden Spiral?

    <p>Clockwise direction (D)</p> Signup and view all the answers

    What is the main purpose of drawing squares of side lengths equal to terms of the Fibonacci sequence?

    <p>To create a connected path (B)</p> Signup and view all the answers

    What is the pattern formed by connecting opposite corners of the squares with arcs?

    <p>A spiral (C)</p> Signup and view all the answers

    What does it mean for a number L to be the limit of a sequence?

    <p>The terms of the sequence get arbitrarily close to L. (D)</p> Signup and view all the answers

    What is the direction in which the squares are drawn to obtain the Golden Spiral?

    <p>Clockwise (B)</p> Signup and view all the answers

    What is the notation for the sequence {0.7, 2/5, π, e, · · · }?

    <p>{an}^∞ n=1 (A)</p> Signup and view all the answers

    What is the name of the aesthetic use that involves drawing squares of side lengths equal to terms of the Fibonacci sequence?

    <p>Golden Spiral (C)</p> Signup and view all the answers

    What is true about the sequence {an}^∞ n=1 = {0.1, 0.01, 0.001, 0.0001, · · · }?

    <p>The sequence converges to 0. (C)</p> Signup and view all the answers

    What is the sequence used to draw the Golden Spiral?

    <p>The Fibonacci sequence (C)</p> Signup and view all the answers

    What does the notation lim an = L mean?

    <p>The sequence an converges to L. (C)</p> Signup and view all the answers

    What is a common characteristic of sequences that have a limit?

    <p>The terms of the sequence get arbitrarily close to a fixed value. (B)</p> Signup and view all the answers

    What is the common factor among the terms of the Fibonacci sequence?

    <p>1 (A)</p> Signup and view all the answers

    What is the notation for the nth element of a sequence?

    <p>an (A)</p> Signup and view all the answers

    What is the relationship between the terms of the Fibonacci sequence?

    <p>Each term is the sum of the previous two terms (C)</p> Signup and view all the answers

    What is the main application of the Fibonacci sequence mentioned in the text?

    <p>Aesthetics (B)</p> Signup and view all the answers

    What is true about the sequence {cn}^∞ n=1 = {1, 2, 3, 4, · · · }?

    <p>The sequence has no limit. (C)</p> Signup and view all the answers

    What is a common assumption about limits?

    <p>The limit of a sequence is assumed to exist. (A)</p> Signup and view all the answers

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