From 902b854656fd1f5e39cfbdb331809c64b5671e4c Mon Sep 17 00:00:00 2001 From: tomit4 Date: Sun, 12 Jul 2026 05:54:11 -0700 Subject: [PATCH] :construction: Setup for 5.8 --- chapter_5/exercises.md | 189 ++++++++++++++++++++++++++++++++ chapter_5/notes.md | 214 +++++++++++++++++++++++++++++++++++++ chapter_5/test_yourself.md | 25 +++++ 3 files changed, 428 insertions(+) diff --git a/chapter_5/exercises.md b/chapter_5/exercises.md index 79f18df..3c110d5 100644 --- a/chapter_5/exercises.md +++ b/chapter_5/exercises.md @@ -10890,3 +10890,192 @@ b. (For students who have studied calculus) Show that if $0 < m < 1$, then $\lim\limits_{m \to \infty}Y_n = \dfrac{E + c}{1 - m}$. Omitted. + +--- + +Page 385 + +**Exercise Set 5.8** + +1. Which of the following are second-order linear homogeneous recurrence + relations with constant coefficients? + +a. $a_k = 2a_{k - 1} - 5a_{k - 2}$ + +b. $b_k = kb_{k - 1} + b_{k - 2}$ + +c. $c_k = 3c_{k - 1} \cdot c_{k - 2}^2$ + +d. $d_k = 3d_{k - 1} + d_{k - 2}$ + +e. $r_k = r_{k - 1} - r_{k - 2} - 2$ + +f. $s_k = 10s_{k - 2}$ + +2. Which of the following are second-order linear homogeneous recurrence + relations with constant coefficients? + +a. $a_k = (k - 1)a_{k - 1} + 2ka_{k - 2}$ + +b. $b_k = -b_{k - 1} + 7b_{k - 2}$ + +c. $c_k = 3c_{k - 1} + 1$ + +d. $d_k = 3d_{k - 1}^2 + d_{k - 2}$ + +e. $r_k = r_{k - 1} + 6r_{k - 3}$ + +f. $s_k = s_{k - 1} + 10s_{k - 2}$ + +3. Let $a_0, a_1, a_2, \dots$ be the sequence defined by the explicit formula + +$$ a_n = C \cdot 2^n + D \quad \text{ for every integer } n \geq 0 $$ + +where $C$ and $D$ are real numbers. + +a. Find $C$ and $D$ so that $a_0 = 1$ and $a_1 = 3$. What is $a_2$ in this case? + +b. Find $C$ and $D$ so that $a_0 = 0$ and $a_1 = 2$. What is $a_2$ in this case? + +4. Let $b_0, b_1, b_2, \dots$ be the sequence defined by the explicit formula + +$$ b_n = C \cdot 3^n + D(-2)^n \quad \text{ for each integer } n \geq 0 $$ + +where $C$ and $D$ are real numbers. + +a. Find $C$ and $D$ so that $b_0 = 0$ and $b_1 = 5$. What is $b_2$ in this case? + +b. Find $C$ and $D$ so that $b_0 = 3$ and $b_1 = 4$. What is $b_2$ in this case? + +5. Let $a_0, a_1, a_2, \dots$ be the sequence defined by the explicit formula + +$$ a_n = C \cdot 2^n + D \quad \text{ for each integer } n \geq 0 $$ + +where $C$ and $D$ are real numbers. Show that for any choice of $C$ and $D$, + +$$ a_k = 3a_{k - 1} - 2a_{k - 2} \quad \text{ for every integer } k \geq 2 $$ + +6. Let $b_0, b_1, b_2, \dots$ be the sequence defined by the explicit formula + +$$ b_n = C \cdot 3^n + D(-2)^n \quad \text{ for every integer } n \geq 0 $$ + +where $C$ and $D$ are real numbers. Show that for any choice of $C$ and $D$, + +$$ b_k = b_{k - 1} + 6b_{k - 2} \quad \text{ for each integer } k \geq 2 $$ + +7. Solve the system of equations in Example 5.8.4 to obtain + +$$ C = \frac{1 + \sqrt{5}}{2\sqrt{5}} \quad \text{ and } \quad D = \frac{-(1 - \sqrt{5})}{2\sqrt{5}} $$ + +In each of 8-10: (a) suppose a sequence of the form +$1, t, t^2, t^3, \dots, t^n, \dots$ where $t \neq 0$, satisfies the given +recurrence relation (but not necessarily the initial conditions), and find all +possible values of $t$: (b) suppose a sequence satisfies the given initial +conditions as well as the recurrence relation, and find an explicit formula for +the sequence. + +8. $a_k = 2a_{k - 1} + 3a_{k - 2}$, for every integer $k \geq 2$ + $a_0 = 1, a_1 = 2$ + +9. $b_k = 7b_{k - 1} - 10b_{k - 2}$, for every integer $k \geq 2$ + $b_0 = 2, b_1, = 2$ + +10. $c_k = c_{k - 1} + 6c_{k - 2}$, for every integer $k \geq 2$ + $c_0 = 0, c_1 = 3$ + +In each of 11-16 suppose a sequence satisfies the given recurrence relation and +initial conditions. Find an explicit formula for the sequence. + +11. $d_k = 4d_{k - 2}$ , for each integer $k \geq 2$ $d_0 = 1, d_1 = -1$ + +12. $e_k = 9e_{k - 1}$, for each integer $k \geq 2$ $e_0 = 0, e_1 = 2$ + +13. $r_k = 2r^{k - 1} - r^{k - 2}$, for each integer $k \geq 2$ + $r_0 = 1, r_1 = 4$ + +14. $s_k = -4s_{k - 1} - 4s_{k - 2}$, for every integer $k \geq 2$ + $s_0 = 0, s_1 = -1$ + +15. $t_k = 6t_{k - 1} - 9t_{k - 2}$, for each integer $k \geq 2$ + $t_0 = 1, t_1= 3$ + +16. $s_k = 2s_{k - 1} + 2s_{k - 2}$, for every integer $k \geq 2$ + $s_0 = 1, s_1 = 3$ + +17. Find an explicit formula for the sequence of exercise 39 in Section 5.6. + +18. Suppose that the sequences $s_0, s_1, s_2, \dots$ and $t_0, t_1, t_2, \dots$ + both satisfy the same second-order linear homogeneous recurrence relation + with constant coefficients: + +$$ s_k = 5s_{k - 1} - 4s_{k - 2} \quad \text{ for each integer } k \geq 2 $$ + +$$ t_k = 5t_{k - 1} - 4t_{k - 2} \quad \text{ for each integer } k \geq 2 $$ + +Show that the sequence $2s_0 + 3t_0, 2s_1 + 3t_1, 2s_2 + 3t_2, \dots$ also +satisfies the same relation. In other words, show that + +$$ 2s_k + 3t_k = 5(2s_{k - 1} + 3t_{k - 1}) - 4(2s_{k - 2} + 3t_{k - 2}) $$ + +for each integer $k \geq 2$. Do _not_ use Lemma 5.8.2. + +19. Show that if $r, s, a_0$, and $a_1$ are numbers with $r \neq s$, then there + exist unique numbers $C$ and $D$ so that + +$$ C + D = a_0 $$ + +$$ Cr + Ds = a_1 $$ + +20. Show that if $r$ is a nonzero real number, $k$ and $m$ are distinct + integers, and $a_k$ and $a_m$ are any real numbers, then there exist unique + real numbers $C$ and $D$ so that + +$$ Cr^k + kDr^k = a_k $$ + +$$ Cr^m + mDr^m = a_m $$ + +21. Prove Theorem 5.8.5 for the case where the values of $C$ and $D$ are + determined by $a_0$ and $a_1$. + +Exercises 22 and 23 are intended for students who are familiar with complex +numbers. + +22. Find an explicit formula for a sequence $a_0, a_1, a_2, \dots$ that + satisfies + +$$ a_k = 2a_{k - 1} - 2a_{k - 1} \quad \text{ for every integer } k \geq 2 $$ + +with initial conditions $a_0 = 1$ and $a_1 = 2$. + +23. Find an explicit formula for a sequence $b_0, b_1, b_2, \dots$ that + satisfies + +$$ b_k = 2b_{k - 1} - 5b_{k - 2} \quad \text{ for each integer } k \geq 2 $$ + +with initial conditions $b_0 = 1$ and $b_1 = 1$. + +24. The numbers $\dfrac{1 + \sqrt{5}}{2}$ and $\dfrac{1 - \sqrt{5}}{2}$ that + appear in the explicit formula for the Fibonacci sequence are related to a + quantity called the _golden ratio_ in Greek mathematics. Consider a + rectangle of length $\phi$ units and height $1$, where $\phi > 1$. + +See page 387 for picture. + +Divide the rectangle into a rectangle and a square as shown in the preceding +diagram. The square is $1$ unit on each side, and the rectangle has sides of +length $1$ and $\phi - 1$. The ancient Greeks considered the outer rectangle to +be perfectly proportioned (saying that the lengths of its sides are in a _golden +ratio_ to each other) if the ratio of the length to the width of the outer +rectangle equals the ratio of the length to the width of the inner rectangle. +That is, if the number $\phi$ satisfies the equation + +$$ \frac{\phi}{1} = \frac{1}{\phi - 1} $$ + +a. Show that if $\phi$ satisfies the equation above, then it also satisfies the +quadratic equation: $t^2 - t - 1 = 0$. + +b. Find the two solutions of $t^2 - t - 1 = 0$ and call them $\phi_1$ and +$\phi_2$. + +c. Express the explicit formula for the Fibonacci sequence in terms of $\phi_1$ +and $\phi_2$. diff --git a/chapter_5/notes.md b/chapter_5/notes.md index 9d11a16..25ab051 100644 --- a/chapter_5/notes.md +++ b/chapter_5/notes.md @@ -1030,3 +1030,217 @@ $$ a_k = ra_{k - 1} \quad \text{ for each integer } k \geq 1 $$ It follows that $$ a_n = a_0r^n \quad \text{ for each integer } n \geq 0 $$ + +--- + +Page 376 + +**Definition** + +A **second-order linear homogeneous recurrence relation with constant +coefficients** is a recurrence relation of the form + +$$ a_k = Aa_{k - 1} + Ba_{k - 2} \quad \text{ for every integer } k \geq \text{ some fixed integer} $$ + +where $A$ and $b$ are fixed real numbers with $B \neq 0$. + +--- + +Page 377 + +**Lemma 5.8.1** + +Let $A$ and $B$ be real numbers. A recurrence relation of the form + +$$ a_k = Aa_{k - 1} + Ba_{k - 2} \quad \text{ for every integer } k \geq 2 $$ + +is satisfied by the sequence + +$$ 1, t, t^2, t^3, \dots, t^n, \dots , $$ + +where $t$ is a nonzero real number, if, and only if, $t$ satisfies the equation + +$$ t^2 - At - B = 0 $$ + +--- + +Page 377 + +**Definition** + +Given a second-order linear homogeneous recurrence relation with constant +coefficients + +$$ a_k = Aa_{k - 1} + Ba_{k - 2} \quad \text{ for every integer } k \geq 2 $$ + +the **characteristic equation of the relation** is + +$$ t^2 - At - B = 0 $$ + +--- + +Page 378 + +**Lemma 5.8.2** + +If $r_0, r_1, r_2, \dots$ and $s_0, s_1, s_2, \dots$ are sequences that satisfy +the same second-order linear homogeneous recurrence relation with constant +coefficients, and if $C$ and $D$ are _any_ numbers, then the sequence +$a_0, a_1, a_2, \dots$ defined by the formula + +$$ a_n = Cr_n +Ds_n \quad \text{ for every integer } n \geq 0 $$ + +also satisfies the same recurrence relation. + +**Proof:** + +Suppose $r_0, r_1, r_2, \dots$, and $s_0, s_1, s_2, \dots$ are sequences that +satisfy the same second-order linear homogeneous recurrence relation with +constant coefficients. In other words, suppose that for some real numbers $A$ +and $B$, + +$$ r_k = Ar_{k - 1} + Br_{k - 2} \quad \text{ and } \quad s_k = As_{k - 1} + Bs_{k - 2} $$ + +for every integer $k \geq 2$. Suppose also that $C$ and $D$ are any numbers. Let +$a_0, a_1, a_2, \dots$ be the sequence defined by + +$$ a_n = Cr_n + Ds_n \quad \text{ for every integer } n \geq 0 $$ + +_[We must show that $a_0, a_1, a_2, \dots$ satisfies the same recurrence +relation as $r_0, r_1, r_2, \dots$ and $s_0, s_1, s_2, \dots$. That is we must +show that $a_k = Aa_{k - 1} + Ba_{k - 2}$, for every integer $k \geq 2$.]_ + +For every integer $k \geq 2$, + +$$ Aa_{k - 1} + Ba_{k - 2} = A(Cr_{k - 1} + Ds_{k - 1}) + B(Cr_{k - 2} + Ds_{k - 2}) $$ + +$$ = C(Ar_{k - 1} + Br{k - 2}) + D(As_{k - 1} + Bs_{k - 2}) $$ + +$$ = Cr_k + Ds_k $$ + +$$ = a_k $$ + +Hence $a_0, a_1, a_2, \dots$ satisfies the same recurrence relation as +$r_0, r_1, r_2, \dots$ and $s_0, s_1, s_2, \dots$ _[as was to be shown]._ + +--- + +Page 380 + +**Theorem 5.8.3 Distinct-Roots Theorem** + +Suppose a sequence $a_0, a_1, a_2, \dots$ satisfies a recurrence relation + +$$ a_k = Aa_{k - 1} + Ba_{k - 2} $$ + +for some real numbers $A$ and $B$ with $B \neq 0$ and every integer $k \geq 2$. +If the characteristic equation + +$$ t^2 - At - B = 0 $$ + +has two distinct roots $r$ and $s$, then $a_0, a_1, a_2, \dots$ is given by the +explicit formula + +$$ a_n = Cr^n + Ds^n $$ + +where $C$ and $D$ are the numbers whose values are determined by the values +$a_0$ and $a_1$. + +**Proof:** + +Suppose that for some real numbers $A$ and $B$, a sequence +$a_0, a_1, a_2, \dots$ satisfies the recurrence relation +$a_k = Aa_{k - 1} + Ba_{k - 2}$, for every integer $k \geq 2$, and suppose the +characteristic equation $t^2 - At - B = 0$ has two distinct roots $r$ and $s$. +We will show that + +$$ \text{for every integer } n \geq 0, \quad a_n = Cr^n + Ds^n $$ + +where $C$ and $D$ are numbers such that + +$$ a_0 = Cr^0 + Ds^0 \quad \text{ and } \quad a_1 = Cr^1 + Ds^1 $$ + +Let $P(n)$ be the equation + +$$ a_n = Cr^n + Ds^n $$ + +We use strong mathematical induction to prove that $P(n)$ is true for each +integer $n \geq 0$. In the basis step, we prove that $P(0)$ and $P(1)$ are true. +We do this because in the inductive step we need the equation to hold for +$n = 0$ and $n = 1$ in order to prove that it holds for $n = 2$. + +_Show that $P(0)$ and $P(1)$ are true:_ + +The truth of $P(0)$ and $P(1)$ is automatic because $C$ and $D$ are exactly +those numbers that make the following equations true: + +$$ a_0 = Cr^0 + Ds^0 \quad \text{ and } \quad a_1 = Cr^1 + Ds^1 $$ + +_Show that for every integer $k \geq 1$, if $P(i)$ is true for each integer $i$ +from $0$ through $k$, then $P(k + 1)$ is also true:_ + +Suppose that $k$ is any integer with $k \geq 1$ and for each integer $i$ from +$0$ through $k$, + +$$ a_i = Cr^i + Ds^i $$ + +We must show that + +$$ a_{k + 1} = Cr^{k + 1} + Ds^{k + 1} $$ + +Now by the inductive hypothesis, + +$$ a_k = Cr^k + Ds^k \quad \text{ and } \quad a_{k - 1} = Cr^{k - 1} + Ds^{k - 1} $$ + +so + +$$ a_{k + 1} = Aa_k + Ba_{k - 1} $$ + +$$ = A(Cr^k + Ds^k) + B(Cr^{k - 1} + Ds^{k - 1}) $$ + +$$ = C(Ar^k + Br^{k - 1}) + D(As^k + Bs^{k - 1}) $$ + +$$ = Cr^{k + 1} + Ds^{k + 1} $$ + +This is what was to be shown. + +_[The reason the last equality follows from Lemma 5.8.1 is that since $r$ and +$s$ satisfy the characteristic equation (5.8.2), the sequences +$r^0, r^1, r^2, \dots$ and $s^0, s^1, s^2, \dots$ satisfy the recurrence +relation (5.8.1).]_ + +--- + +Page 384 + +**Lemma 5.8.4** + +Let $A$ and $B$ be real numbers and suppose the characteristic equation + +$$ t^2 - At - B = 0 $$ + +has a single root $r$. Then the sequences $1, r^1, r^2, r^3, \dots, r^n, \dots$ +and $0, r, 2r^2, 3r^3, \dots, nr^n, \dots$ both satisfy the recurrence relation + +$$ a_k = Aa_{k - 1} + Ba_{k - 2} $$ + +for each integer $k \geq 2$. + +--- + +Page 384 + +**Theorem 5.8.5 Single-Root Theorem** + +Suppose a sequence $a_0, a_1, a_2, \dots$ satisfies a recurrence relation + +$$ a_k = Aa_{k - 1} + Ba_{k - 2} $$ + +for some real numbers $A$ and $B$ with $B \neq 0$ and for every integer +$k \geq 2$. If the characteristic equation $t^2 - At - B = 0$ has a single +(real) root $r$, then $a_0, a_1, a_2, \dots$ is given by the explicit formula + +$$ a_n = Cr^n + Dnr^n $$ + +where $C$ and $D$ are the real numbers whose values are determined by the values +of $a_0$ and any other known value of the sequence. diff --git a/chapter_5/test_yourself.md b/chapter_5/test_yourself.md index e66f556..1243935 100644 --- a/chapter_5/test_yourself.md +++ b/chapter_5/test_yourself.md @@ -218,3 +218,28 @@ $a_k = ra_{k - 1}$; $a_n = r^na_0$ by iteration, its correctness can be checked by _____. mathematical induction + +--- + +Page 385 + +**Test Yourself** + +1. A second-order linear homogeneous recurrence relation with constant + coefficients is a recurrence relation of the form _____ for every integer + $k \geq$ _____, where _____. + +2. Given a recurrence relation of the form $a_k = Aa_{k - 1} + Ba_{k - 2}$ for + every integer $k \geq 2$, the characteristic equation of the relation is + _____. + +3. If a sequence $a_1, a_2, a_3, \dots$ is defined by a second-order linear + homogeneous recurrence relation with constant coefficients and the + characteristic equation for the relation has two distinct roots $r$ and $s$ + (which could be complex numbers), then the sequence is given by an explicit + formula of the form _____. + +4. If a sequence $a_1, a_2, a_3, \dots$ is defined by a second-order linear + homogeneous recurrence relation with constant coefficients and the + characteristic equation for the relation has only a single root $r$, then the + sequence is given by an explicit formula of the form _____.