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