Excursions in Modern Mathematics

Consider the following sequence of equations involving Fibonacci numbers.
$$\begin{array}{l}1+2=3 \\1+2+5=8 \\1+2+5+13=21 \\1+2+5+13+34=55\end{array}$$
(a) Write down a reasonable choice for the fifth equation in this sequence.
(b) Find the subscript that will make the following equation true.
$$F_{1}+F_{3}+F_{5}+\cdots+F_{21}=F_{?}$$
(c) Find the subscript that will make the following equation true (assume $N$ is odd).
$$F_{1}+F_{3}+F_{5}+\cdots+F_{N}=F$$

Julian Wong

Numerade Educator

07:19

Problem 14

Consider the following sequence of equations involving Fibonacci numbers.
$$\begin{array}{l}2(2)-3=1 \\2(3)-5=1 \\2(5)-8=2 \\2(8)-13=3 \\\vdots\end{array}$$
(a) Write down a reasonable choice for the fifth equation in this sequence.
(b) Find the subscript that will make the following equation true
$$2\left(F_{?}\right)-F_{15}=F_{12}$$
(c) Find the subscript that will make the following equation true.
$$2\left(F_{N+2}\right)-F_{N+3}=F_{?}$$

Kevin Shryock

Numerade Educator

Problem 15

Fact: If we make a list of any four consecutive Fibonacci numbers, the first one times the fourth one is always equal to the third one squared minus the second one squared.
(a) Verify this fact for the list $F_{8}, F_{9}, F_{10}, F_{11}$.
(b) Using the list $F_{N}, F_{N+1}, F_{N+2}, F_{N+3},$ write this fact as a mathematical formula.

Prathan Jarupoonphol

Numerade Educator

Problem 16

Fact: If we make a list of any 10 consecutive Fibonacci numbers, the sum of all these numbers divided by 11 is always equal to the seventh number on the list.
(a) Verify this fact for the list $F_{1}, F_{2}, \ldots, F_{10}$
(b) Using the list $F_{N}, F_{N+1}, \ldots, F_{N+9},$ write this fact as a mathematical formula.

Prathan Jarupoonphol

Numerade Educator

04:03

Problem 17

Express each of the following as a single Fibonacci number.
(a) $\quad F_{N+1}+F_{N+2}=$
(b) $\quad F_{N}-F_{N-2}=$
(c) $\quad F_{N}+F_{N+1}+F_{N+3}+F_{N+5}=$

Prathan Jarupoonphol

Numerade Educator

04:03

Problem 18

Express each of the following as a single Fibonacci number.
(a) $\quad F_{N-2}+F_{N-3}=$
(b) $\quad F_{N+2}-F_{N}=$
(c) $\quad F_{N-3}+F_{N-2}+F_{N}+F_{N+2}=$

Prathan Jarupoonphol

Numerade Educator

04:26

Problem 19

Express each of the following as a ratio of two Fibonacci numbers.
(a) $1+\frac{F_{N}}{F_{N-1}}=$
(b) $\frac{F_{N-1}}{F_{N}}-1=$

Runpeng Li

Numerade Educator

04:26

Problem 20

Express each of the following as a ratio of two Fibonacci numbers.
(a) $1+\frac{F_{N-1}}{F_{N}}=$
(b) $1-\frac{F_{N}}{F_{N-2}}=$

Runpeng Li

Numerade Educator

Problem 21

Consider the Fibonacci-like sequence $5,5,10,15,25,40, \ldots$ and let $A_{N}$ denote the $N$ th term of the sequence.
(a) Find $A_{10}$.
(b) Given that $F_{25}=75,025,$ find $A_{25}$
(c) Express $A_{N}$ in terms of $F_{N}$.

Julian Wong

Numerade Educator

Problem 22

Consider the Fibonacci-like sequence $2,4,6,10,16,26, \ldots,$ and let $B_{N}$ denote the $N$th term of the sequence.
(a) Find $B_{9}$.
(b) Given that $F_{20}=6765,$ find $B_{19}$
(c) Express $B_{N}$ in terms of $F_{N+1}$.

Julian Wong

Numerade Educator

07:31

Problem 23

Consider the Fibonacci-like sequence 1,3,4,7,11,18,29 $47, \ldots,$ and let $L_{N}$ denote the $N$ th term of the sequence. (Note: This sequence is called the Lucas sequence, and the terms of the sequence are called the Lucas numbers.)
(a) Find $L_{12}$.
(b) The Lucas numbers are related to the Fibonacci numbers by the formula $L_{N}=2 F_{N+1}-F_{N}$. Verify that this formula is true for $N=1,2,3,$ and 4
(c) Given that $F_{20}=6765$ and $F_{21}=10,946,$ find $L_{20}$

Chris Trentman

Numerade Educator

Problem 24

Consider the Fibonacci-like sequence $1,4,5,9,14,23,37, \ldots,$ and let $T_{N}$ denote the $N$ th term of the sequence.
(a) Find $T_{12}$.
(b) The numbers in this sequence are related to the Fibonacci numbers by the formula $T_{N}=3 F_{N+1}-2 F_{N}$. Verify that this formula is true for $N=1,2,3,$ and 4
(c) Given that $F_{20}=6765$ and $F_{21}=10,946,$ find $T_{20}$.

Julian Wong

Numerade Educator

Problem 25

Consider the quadratic equation $x^{2}=x+1$
(a) Use the quadratic formula to find the two solutions of the equation. Give the value of each solution rounded to five decimal places.
(b) Find the sum of the two solutions in (a).
(c) Explain why the decimal part has to be exactly the same in both solutions.

Cory Kuzinski

Numerade Educator

Problem 26

Consider the quadratic equation $x^{2}=3 x+1$
(a) Use the quadratic formula to find the two solutions of the equation. Give the value of each solution rounded to five decimal places.
(b) Find the sum of the two solutions in (a).
(c) Explain why the decimal part has to be exactly the same in both solutions.

Cory Kuzinski

Numerade Educator

01:35

Problem 27

Consider the quadratic equation $3 x^{2}=8 x+5$.
(a) Use the quadratic formula to find the two solutions of the equation. Give the value of each solution rounded to five decimal places.
(b) Find the sum of the two solutions found in (a).

Teresa Liang

Numerade Educator

04:20

Problem 28

Consider the quadratic equation $8 x^{2}=5 x+2$
(a) Use the quadratic formula to find the two solutions of the equation. Give the value of each solution rounded to five decimal places.
(b) Find the sum of the two solutions found in (a).

Laurie Buchanan

Numerade Educator

00:18

Problem 29

Consider the quadratic equation $55 x^{2}=34 x+21$
(a) Without using the quadratic formula, show that $x=1$ is one of the two solutions of the equation.
(b) Without using the quadratic formula, find the second solution of the equation. (Hint: The sum of the two solutions of $a x^{2}+b x+c=0$ is given by $-b / a$.)

Amy Jiang

Numerade Educator

Problem 30

Consider the quadratic equation $89 x^{2}=55 x+34$
(a) Without using the quadratic formula, show that $x=1$ is one of the two solutions of the equation.
(b) Without using the quadratic formula, find the second solution of the equation. (Hint: The sum of the two solutions of $a x^{2}+b x+c=0$ is given by $-b / a$.)

Cory Kuzinski

Numerade Educator

01:39

Problem 31

Consider the quadratic equation $21 x^{2}=34 x+55 .$
(a) Without using the quadratic formula, show that $x=-1$ is one of the two solutions of the equation.
(b) Without using the quadratic formula, find the second solution of the equation. (Hint: The sum of the two solutions of $a x^{2}+b x+c=0$ is given by $-b / a$.)

Nick Johnson

Numerade Educator

Problem 32

Consider the quadratic equation $34 x^{2}=55 x+89 .$
(a) Without using the quadratic formula, show that $x=-1$ is one of the two solutions of the equation.
(b) Without using the quadratic formula, find the second solution of the equation. (Hint: The sum of the two solutions of $a x^{2}+b x+c=0$ is given by $-b / a$.)

Cory Kuzinski

Numerade Educator

03:59

Problem 33

Consider the quadratic equation $\left(F_{N}\right) x^{2}=\left(F_{N-1}\right) x+F_{N-2},$ where $F_{N-2}, F_{N-1},$ and $F_{N}$ are consecutive Fibonacci numbers.
(a) Show that $x=1$ is one of the two solutions of the equation. [Hint: Try Exercises 29 (a) or 30 (a) first.]
(b) Find the second solution of the equation expressed in terms of Fibonacci numbers. [Hint: Try Exercises 29 (b) or $30($ b) first. $]$

Angela Guo

Numerade Educator

03:59

Problem 34

Consider the quadratic equation $\left(F_{N-2}\right) x^{2}=\left(F_{N-1}\right) x+F_{N}$, where $F_{N-2}, F_{N-1},$ and $F_{N}$ are consecutive Fibonacci numbers
(a) Show that $x=-1$ is one of the two solutions of the equation. [Hint: Try Exercises $31($ a) or $32($ a) first.
(b) Find the second solution of the equation expressed in terms of Fibonacci numbers. [Hint: Try Exercises $31(\mathrm{~b})$ or $32(\mathrm{~b})$ first.

Angela Guo

Numerade Educator

01:19

Problem 35

The number $\frac{1}{\phi}$ is the reciprocal of the golden ratio.
(a) Using a calculator, compute $\frac{1}{\phi}$ to 10 decimal places.
(b) Explain why $\frac{1}{\phi}$ has exactly the same decimal part as $\phi$.

Ankit Gupta

Numerade Educator

00:59

Problem 36

The square of the golden ratio is the irrational number
$$\phi^{2}=\left(\frac{1+\sqrt{5}}{2}\right)^{2}=\frac{3+\sqrt{5}}{2}$$
(a) Using a calculator, compute $\phi^{2}$ to 10 decimal places.
(b) Explain why $\phi^{2}$ has exactly the same decimal part as $\phi$.

Matt Gibson

Numerade Educator

01:11

Problem 37

Given that $F_{499} \approx 8.6168 \times 10^{103}$, find an approximate value for $F_{500}$ in scientific notation. (Hint: $F_{N} / F_{N-1} \approx \phi .$

Julie Silva

Numerade Educator

00:56

Problem 38

Given that $F_{1002} \approx 1.138 \times 10^{209}$, find an approximate value for $F_{1000}$ in scientific notation. (Hint: $F_{N} / F_{N-1} \approx \phi .$ )

Suman Saurav Thakur

Numerade Educator

18:48

Problem 39

The Fibonacci sequence of order 2 is the sequence of numbers $1,2,5,12,29,70, \ldots$ Each term in this sequence (from the third term on) equals two times the term before it plus the term two places before it; in other words, $A_{N}=2 A_{N-1}+A_{N-2}(N \geq 3)$
(a) Compute $A_{7}$.
(b) Use your calculator to compute to five decimal places the ratio $A_{7} / A_{6}$
(c) Use your calculator to compute to five decimal places the ratio $A_{11} / A_{10}$ -
(d) Guess the value (to five decimal places) of the ratio $A_{N} / A_{N-1}$ when $N>11$

Tara Stewart

Numerade Educator

00:54

Problem 40

The Fibonacci sequence of order 3 is the sequence of numbers $1,3,10,33,109, \ldots$ Each term in this sequence (from the third term on) equals three times the term before it plus the term two places before it; in other words, $A_{N}=3 A_{N-1}+A_{N-2}(N \geq 3)$
(a) Compute $A_{6}$.
(b) Use your calculator to compute to five decimal places the ratio $A_{6} / A_{5}$
(c) Guess the value (to five decimal places) of the ratio $A_{N} / A_{N-1}$ when $N>6$

Lily An

Numerade Educator

01:21

Problem 41

$R$ and $R^{\prime}$ are similar rectangles. Suppose that the width of $R$ is $a$ and the width of $R^{\prime}$ is $3 a$.
(a) If the perimeter of $R$ is 41.5 in., what is the perimeter of $R^{\prime \prime}$
(b) If the area of $R$ is $105 \mathrm{sq} .$ in., what is the area of $R^{\prime} ?$

Dharmendra Jain

Numerade Educator

05:13

Problem 42

$O$ and $O^{\prime}$ are similar O-rings. The inner radius of $O$ is $5 \mathrm{ft}$ and the inner radius of $O^{\prime}$ is $15 \mathrm{ft}$.
(a) If the circumference of the outer circle of $O$ is $14 \pi \mathrm{ft}$ what is the circumference of the outer circle of $O^{\prime} ?$
(b) Suppose that it takes 1.5 gallons of paint to paint the O-ring $O$. If the paint is used at the same rate, how much paint is needed to paint the O-ring $O^{\prime} ?$

Derrick Hanson

Numerade Educator

12:01

Problem 43

Triangles $T$ and $T^{\prime}$ shown in Fig. 23 are similar triangles.
(a) If the perimeter of $T$ is 13 in., what is the perimeter of $T^{\prime}$ (in meters)?
(b) If the area of $T$ is 20 sq. in., what is the area of $T^{\prime}$ (in square meters)?

Marie Lucie Gauthe

Numerade Educator

01:45

Problem 44

Polygons $P$ and $P^{\prime}$ shown in Fig. 24 are similar polygons.
(a) If the perimeter of $P$ is $10,$ what is the perimeter of $P^{\prime} ?$
(b) If the area of $P$ is 30 , what is the area of $P^{\prime}$ ?

Glenn Degamon

Numerade Educator

01:16

Problem 45

Find the value of $c$ so that the shaded rectangle in Fig. 25 is a gnomon to the white 3 by 9 rectangle. (Figure is not drawn to scale.)

Sherrie Fenner

Numerade Educator

01:12

Problem 46

Find the value of $x$ so that the shaded figure in Fig. 26 is a gnomon to the white rectangle. (Figure is not drawn to scale.

Elisa Ma

Numerade Educator

01:12

Problem 47

Find the value of $x$ so that the shaded figure in Fig. 27 is a gnomon to the white rectangle. (Figure is not drawn to scale.)

Elisa Ma

Numerade Educator

01:27

Problem 48

Find the value of $x$ so that the shaded figure in Fig. 28 is a gnomon to the white rectangle. (Figure is not drawn to scale.)

Grace Muhihu

Numerade Educator

01:49

Problem 49

Rectangle $A$ is 10 by $20 .$ Rectangle $B$ is a gnomon to rectangle $A$. What are the dimensions of rectangle $B$?

Teresa Fuston

Numerade Educator

01:27

Problem 50

Find the value of $x$ so that the shaded frame in Fig. 29 is a gnomon to the white $x$ by 8 rectangle. (Figure is not drawn to scale.)

Grace Muhihu

Numerade Educator

01:00

Problem 51

In Fig. 30 triangle $B C A$ is a $36-36-108$ triangle with sides of length $\phi$ and $1 .$ Suppose that triangle $A C D$ is a gnomon to triangle $B C A$
(a) Find the measure of the angles of triangle $A C D$.
(b) Find the length of the three sides of triangle $A C D$.

Doruk Isik

Numerade Educator

02:40

Problem 52

Find the values of $x$ and $y$ so that in Fig. 31 the shaded triangle is a gnomon to the white triangle $A B C$.

Thomas Emment

Numerade Educator

03:27

Problem 53

Find the values of $x$ and $y$ so that in Fig. 32 the shaded figure is a gnomon to the white triangle.

Ankit Gupta

Numerade Educator

00:53

Problem 54

Find the values of $x$ and $y$ so that in Fig. 33 the shaded triangle is a gnomon to the white triangle.

Lucas Finney

Numerade Educator

01:02

Problem 55

Consider the sequence of ratios $\frac{F_{N}}{F_{N+1}}$.
(a) Using a calculator compute the first 14 terms of this sequence in decimal form (rounded to six decimal places when needed).
(b) Explain why $\left(\frac{F_{N}}{F_{N+1}}\right) \rightarrow \phi-1$ (i.e., as $N$ gets larger and larger, the ratios $\frac{F_{N}}{F_{N+1}}$ get closer and closer to $\left.\phi-1\right)$.

Amit Srivastava

Numerade Educator

01:02

Problem 56

Consider the sequence of ratios $\frac{F_{N+2}}{F_{N}}$
(a) Using a calculator compute the first 15 terms of this sequence in decimal form (rounded to six decimal places when needed).
(b) Explain why $\left(\frac{F_{N+2}}{F_{N}}\right) \rightarrow \phi+1$ (i.e., as $N$ gets larger and larger, the ratios $\frac{F_{N+2}}{F_{N}}$ get closer and closer to $\phi+1$ ).

Amit Srivastava

Numerade Educator

02:04

Problem 57

Consider the sequence $T$ given by the following recursive definition: $T_{N+1}=1+\frac{1}{T_{N}},$ and $T_{1}=1$
(a) Find the first six terms of the sequence, and leave the terms in fractional form.
(b) Explain why $T_{N} \rightarrow \phi$ (i.e., as $N$ gets larger and larger, $T_{N}$ gets closer and closer to $\phi$ ).

Vysakh M

Numerade Educator

02:19

Problem 58

Consider the sequence $U$ given by the following recursive definition: $U_{N+1}=\frac{1}{1+U_{N}},$ and $U_{1}=1 .$ As $N$ gets larger and larger, the terms of this sequence get closer and closer to some number. Give the number expressed in terms of the golden ratio $\phi .$

Darshan Maheshwari

Numerade Educator

07:31

Problem 59

The Lucas sequence is the Fibonacci-like sequence $1,3,4,7,11,18,29,47, \ldots$ (first introduced in Exercise 23 ). The numbers in the Lucas sequence are called the Lucas numbers, and we will use $L_{N}$ to denote the $N$ th Lucas number. The Lucas numbers satisfy the recursive rule $L_{N}=L_{N-1}+L_{N-2}$ (just like the Fibonacci numbers), but start with the initial values $L_{1}=1, L_{2}=3 .$
(a) Show that the Lucas numbers are related to the Fibonacci numbers by the formula $L_{N}=2 F_{N+1}-F_{N}$ [Hint: Let $K_{N}=2 F_{N+1}-F_{N},$ and show that the numbers $K_{N}$ satisfy exactly the same definition as the Lucas numbers (same initial values and same recursive rule).
(b) Show that $\left(\frac{L_{N+1}}{L_{N}}\right) \rightarrow \phi .[$ Hint: Use (a) combined with the fact that $\left.\left(\frac{F_{N+1}}{F_{N}}\right) \rightarrow \phi .\right]$

Chris Trentman

Numerade Educator

05:18

Problem 60

(a) Explain what happens to the values of $\left(\frac{1-\sqrt{5}}{2}\right)^{N}$ as $N$ gets larger. (Hint: Get a calculator and experiment with $N=6,7,8, \ldots$ until you get the picture.)
(b) Explain why $F_{N} \rightarrow \frac{\phi^{N}}{\sqrt{5}}$. [Hint: Use (a) and the original Binet's formula.
(c) Using (b), explain why $\left(\frac{F_{N+1}}{F_{N}}\right) \rightarrow \phi$.

Gokul R Nair

Numerade Educator

00:41

Problem 61

Explain why the only even Fibonacci numbers are those having a subscript that is a multiple of $3 .$

James Kiss

Numerade Educator

01:33

Problem 62

Show that $F_{N+1}^{2}-F_{N}^{2}=\left(F_{N-1}\right)\left(F_{N+2}\right)$.

Aman Gupta

Numerade Educator

00:21

Problem 63

Explain why the shaded figure in Fig. 34 cannot have a square gnomon.

Amy Jiang

Numerade Educator

03:27

Problem 64

Find the values of $x$ and $y$ so that in Fig. 35 the shaded triangle is a gnomon to the white triangle.

Ankit Gupta

Numerade Educator

02:36

Problem 65

Let $A B C D$ be an arbitrary rectangle as shown in Fig. $36 .$ Let $A E$ be perpendicular to the diagonal $B D$ and $E F$ perpendicular to $A B$ as shown. Show that the rectangle $B C E F$ is a gnomon to the rectangle $A D E F$.

Christopher Stanley

Numerade Educator

02:45

Problem 66

In Fig. 37 triangle $B C D$ is a $72-72-36$ triangle with base of length 1 and longer side of length $x$. (Using this choice of values, the ratio of the longer side to the shorter side is $x / 1=x .)$
(a) Show that $x=\phi .$ (Hint: Triangle $A C B$ is similar to triangle $B C D .)$
(b) What are the interior angles of triangle $D A B ?$
(c) Show that in the isosceles triangle $D A B,$ the ratio of the longer to the shorter side is also $\phi$.

Julian Wong

Numerade Educator

04:35

Problem 67

Show that each of the diagonals of the regular pentagon shown in Fig. 38 has length $\phi .$

Harmender Singh Yadav

Numerade Educator

02:01

Problem 68

(a) A regular decagon ( 10 sides) is inscribed in a circle of radius $1 .$ Find the perimeter in terms of $\phi .$
(b) Repeat (a) with radius $r$. Find the perimeter in terms of $\phi$ and $r$

Wendi Zhao

Numerade Educator

01:05

Problem 69

A generic Fibonacci-like sequence has the form $a, b, b+a, 2 b+a, 3 b+2 a, 5 b+$ $3 a, \ldots$ (i.e., the sequence starts with two arbitrary numbers $a$ and $b$ and after that each term of the sequence is the sum of the two previous terms). Let $G_{N}$ denote the $N$ th term of this sequence.
(a) Show that generic Fibonacci-like numbers are related to the Fibonacci numbers by the formula $G_{N}=$ $b F_{N-1}+a F_{N-2 \cdot}$ [Hint: Try Exercise 59 (a) first.]
(b) Show that $\left(G_{N+1} / G_{N}\right) \rightarrow \phi .$ [Hint: Try Exercise 59 (b) first.]

Nick Johnson

Numerade Educator

01:45

Problem 70

You are designing a straight path $2 \mathrm{ft}$ wide using rectangular paving stones with dimensions $1 \mathrm{ft}$ by $2 \mathrm{ft}$. How many different designs are possible for a path of length
(a) $4 \mathrm{ft}$ ?
(b) $8 \mathrm{ft}$ ?
(c) $N \mathrm{ft}$ ?

Heather Zimmers

Numerade Educator

02:56

Problem 71

Show that $F_{1}+F_{2}+F_{3}+\cdots+F_{N}=F_{N+2}-1$.

Charles Carter

Numerade Educator

Problem 72

Show that $F_{1}+F_{3}+F_{5}+\cdots+F_{N}=F_{N+1}$. (Note that on the left side of the equation we are adding the Fibonacci numbers with odd subscripts up to $N$.)

Prathan Jarupoonphol

Numerade Educator