🚧 Setup for 5.6
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@ -7141,3 +7141,332 @@ the Euclidean algorithm, find integers $u$ and $v$ so that
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$\text{gcd}(330, 156) = 330u + 156v$.
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Omitted.
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---
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**Exercise Set 5.6**
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Page 360
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Find the first four terms of each of the recursively defined sequences in 1-8.
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1. $a_k = 2a_{k - 1} + k$, for every integer $k \geq 2$ $a_1 = 1$
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2. $b_k = b_{k - 1} + 3_k$, for every integer $k \geq 2$ $b_1 = 1$
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3. $c_k = k(c_{k - 1})^2$, for every integer $k \geq 1$ $c_0 = 1$
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4. $d_k = k(d_{k - 1})^2$, for every integer $k \geq 1$ $d_0 = 3$
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5. $s_k = s_{k - 1} + 2s_{k - 2}$, for every integer $k \geq 2$, $s_0 = 1$,
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$s_1 = 1$
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6. $t_k = t_{k - 1} + 2t_{k - 2}$, for every integer $k \geq 2$
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$t_0 = -1, t_1 = 2$
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7. $u_k = ku_{k - 1} - u_{k - 2}$, for every integer $k \geq 3$
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$u_1 = 1, u_2 = 1$
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8. $v_k = v_{k - 1} + v_{k - 2} + 1$, for every integer $k \geq 3$
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$v_1 = 1, v_2 = 3$
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9. Let $a_0, a_1, a_2, \dots$ be defined by the formula $a_n = 3n + 1$, for
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every integer $n \geq 0$. Show that this sequence satisfies the recurrence
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relation $a_k = a_{k - 1} + 3$, for every integer $k \geq 1$.
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10. let $b_0, b_1, b_2, \dots$ be defined by the formula $b_n = 4^n$, for every
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integer $n \geq 0$. Show that this sequence satisfies the recurrence
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relation $b_k = 4b_{k - 1}$, for every integer $k \geq 1$.
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11. Let $c_0, c_1, c_2, \dots$ be defined by the formula $c_n = 2^n - 1$ for
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every integer $n \geq 0$. Show that this sequence satisfies the recurrence
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relation $c_k = 2c_{k - 1} + 1$ for every integer $k \geq 1$.
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12. Let $s_0, s_1, s_2, \dots$ be defined by the formula
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$s_n = \dfrac{(-1)^n}{n!}$ for every integer $n \geq 0$. Show that this
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sequence satisfies the following recurrence relation for every integer
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$k \geq 1$:
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$$ s_k = \frac{-s_{k - 1}}{k} $$
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13. Let $t_0, t_1, t_2, \dots$ be defined by the formula $t_n = 2 + n$ for every
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integer $n \geq 0$. Show that this sequence satisfies the following
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recurrence relation for every integer $k \geq 2$:
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$$ t_k = 2t_{k - 1} - t_{k - 2} $$
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14. Let $d_0, d_1, d_2, \dots$ be defined by the formula $d_n = 3^n - 2^n$ for
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every integer $n \geq 0$. Show that this sequence satisfies the following
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recurrence relation for every integer $k \geq 2$:
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$$ d_k = 5d_{k - 1} - 6d_{k - 2} $$
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15. For the sequence of Catalan numbers defined in Example 5.6.4, prove that for
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each integer $n \geq 1$,
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$$ C_n = \frac{1}{4n + 2}\binom{2n + 2}{n + 1}$$
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16. Use the recurrence relation and values for the Tower of Hanoi sequence
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$m_1, m_2, m_3, \dots$ discussed in Example 5.6.5 to compute $m_7$ and
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$m_8$.
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17. _Tower of Hanoi with Adjacency Requirement:_
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Suppose that in addition to the requirement that they never move a larger disk
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on top of a smaller one, the priests who move the disks of the Tower of Hanoi
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are also allowed only to move disks one by one from one pole to an _adjacent_
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pole. Assume poles $A$ and $C$ are at the two ends of the row and pole $B$ is in
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the middle. Let
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$$ a_n = \left[\text{the minimum number of moves needed to transfer a tower of } n \text{ disks from pole } A \text{ to pole } C \right] $$
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a. Find $a_1, a_2$, and $a_3$.
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b. Find $a_4$.
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c. Find a recurrence relation for $a_1, a_2, a_3, \dots$. Justify your answer.
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18. _Tower of Hanoi with Adjacency Requirement:_
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Suppose the same situation as in exercise 17. Let
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$$ b_n = \left[\text{the minimum number of moves needed to transfer a tower of } n \text{ disks from pole } A \text{ to pole } B \right] $$
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a. Find $b_1, b_2$, and $b_3$.
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b. Find $b_4$.
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c. Show that $b_k = a_{k - 1} + 1 + b_{k - 1}$ for each integer $k \geq 2$,
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where $a_1, a_2, a_3, \dots$ is the sequence defined in exercise 17.
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d. Show that $b_k \leq 3b_{k - 1} + 1$ for each integer $k \geq 2$.
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e. Show that $b_k = 3b_{k - 1} + 1$ for each integer $k \geq 2$.
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19. _Four-Pole Tower of Hanoi:_
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Suppose that the Tower of Hanoi problem has four poles in a row instead of
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three. Disks can be transferred one by one from one pole to any other pole, but
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at no time may a larger disk be placed on top of a smaller disk. Let $s_n$ be
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the minimum number of moves needed to transfer the entire tower of $n$ disks
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from the left-most to the right-most pole.
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a. Find $s_1, s_2$, and $s_3$.
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b. Find $s_4$.
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c. Show that $s_k \leq 2s_{k - 2} + 3$ for every integer $k \geq 3$.
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20. _Tower of Hanoi Poles in a Circle:_
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Suppose that instead of being lined up in a row, the three poles for the
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original Tower of Hanoi are placed in a circle. The monks move the disks one by
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one from one pole to another, but they may only move disks one over in a
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clockwise direction and they may never move a larger disk on top of a smaller
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one. Let $c_n$ be the minimum number of moves needed to transfer a pile of $n$
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disks from one pole to the next adjacent pole in the clockwise direction.
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a. Justify the inequality $c_k \leq 4c_{k - 1} + 1$ for each integer $k \geq 2$.
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b. The expression $4c_{k - 1} + 1$ is not the minimum number of moves needed to
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transfer a pile of $k$ disks from one pole to another. Explain, for example, why
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$c_3 \neq 4c_2 + 1$.
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21. _Double Tower of Hanoi:_
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In this variation of the Tower of Hanoi there are three poles in a row and $2n$
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disks, two each of $n$ different sizes, where $n$ is any positive integer.
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Initially one of the poles contains all the disks placed on top of each other in
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pairs of decreasing size. Disks are transferred one by one from one pole to
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another, but at no time may a larger disk be placed on top of a smaller disk.
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However, a disk may be placed on top of one of the same size. Let $t_n$ be the
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minimum number of moves needed to transfer a tower of $2n$ disks from one pole
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to another.
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a. Find $t_1$ and $t_2$.
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b. Find $t_3$.
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c. Find a recurrence relation for $t_1, t_2, t_3, \dots$.
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22. _Fibonacci Variation:_
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A single pair of rabbits (male and female) is born at the beginning of a year.
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Assume the following conditions (which are somewhat more realistic than
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Fibonacci's):
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(1) Rabbit pairs are not fertile during their first months of life but
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thereafter give birth to four new male/female pairs at the end of every month.
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(2) No rabbits die.
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a. Let $r_n = \text{ the number of rabbits alive at the end of month } n$, for
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each integer $n \geq 1$, and let $r_0 = 1$. Find a recurrence relation for
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$r_0, r_1, r_2, \dots$. Justify your answer.
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b. Compute $r_0, r_1, r_2, r_3, r_4, r_5$, and $r_6$.
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c. How many rabbits will there be at the end of the year?
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23. _Fibonacci Variation:_
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A single pair of rabbits (male and female) is born at the beginning of a year.
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Assume the following conditions:
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(1) Rabbit pairs are not fertile during their first _two_ months of life but
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thereafter give birth to three new male/female pairs at the end of every month.
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(2) No rabbits die.
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a. Let
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$s_n = \text{ the number of pairs of rabbits alive at the end of month } n$, for
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each integer $n \geq 1$, and let $s_0 = 1$. Find a recurrence relation for
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$s_0, s_1, s_2, \dots$. Justify your answer.
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b. Compute $s_0, s_1, s_2, s_3, s_4$, and $s_5$.
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c. How many rabbits will there be at the end of the year?
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In 24-34, $F_0, F_1, F_2, \dots$ is the Fibonacci sequence.
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24. Use the recurrence relation and values for $F_0, F_1, F_2, \dots$ given in
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Example 5.6.6 to compute $F_{13}$ and $F_{14}$.
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25. The Fibonacci sequence satisfies the recurrence relation
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$F_k = F_{k - 1} + F_{k - 2}$, for every integer $k \geq 2$.
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a. Explain why the following is true:
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$$ F_{k + 1} = F_k + F_{k - 1} \text{ for each integer } k \geq 1 $$
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b. Write an equation expressing $F_{k + 2}$ in terms of $F_{k + 1}$ and $F_k$.
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c. Write an equation expressing $F_{k + 3}$ in terms of $F_{k + 2}$ and
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$F_{k + 1}$.
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26. Prove that $F_k = 3F_{k - 3} + 2F_{k - 4}$ for every integer $k \geq 4$.
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27. Prove that $F_k^2 - F_{k - 1}^2 = F_kF_{k + 1} - F_{k - 1}F_{k + 1}$, for
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every integer $k \geq 1$.
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28. Prove that $F_{k + 1}^2 - F_k^2 - F_{k - 1}^2 = 2F_kF_{k - 1}$, for each
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integer $k \geq 1$.
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29. Prove that $F_{k + 1}^2 - F_k^2 = F_{k - 1}F_{k + 2}$, for every integer
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$k \geq 1$.
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30. Use mathematical induction to prove that for each integer $n \geq 0$,
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$F_{n + 2}F_n - F_{n + 1}^2 = (-1)^n$.
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31. Use strong mathematical induction to prove that $F_n < 2^n$ for every
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integer $n \geq 1$.
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32. Prove that for each integer $n \geq 0$, $\text{gcd}(F_{n + 1}, F_n) = 1$.
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(The definition of $\text{gcd}$ is given in Section 4.10.)
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33. It turns out that the Fibonacci sequence satisfies the following explicit
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formula: For every integer $F_n \geq 0$,
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$$ F_n = \frac{1}{\sqrt{5}}\left[\left(\frac{1 + \sqrt{5}}{2}\right)^{n + 1} - \left(\frac{1 - \sqrt{5}}{2}\right)^{n + 1}\right] $$
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Verify that the sequence defined by this formula satisfies the recurrence
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relation $F_k = F_{k - 1} + F_{k - 2}$ for every integer $k \geq 2$.
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34. (For students who have studied calculus) Find
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$\lim\limits_{n \to \infty}\left(\dfrac{F_{n + 1}}{F_n}\right)$, assuming
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that the limit exists.
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35. (For students who have studied calculus) Prove that
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$\lim\limits_{n \to \infty}\left(\dfrac{F_{n + 1}}{F_n}\right)$ exists.
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36. (For students who have studied calculus) Define $x_0, x_1, x_2, \dots$ as
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follows:
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$$ x_k = \sqrt{2 + x_{k - 1}} \quad \text{ for each integer } k \geq 1 $$
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$$ x_0 = 0 $$
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Find $\lim\limits_{n \to \infty}x_n$. (Assume that the limit exists.)
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37. _Compound Interest:_
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Suppose a certain amount of money is deposited in an account paying 4% annual
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interest compounded quarterly. For each positive integer $n$, let
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$R_n = \text{ the amount on deposit at the end of the }$ $n$<sup>th</sup>
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quarter, assuming no additional deposits or withdrawals, and let $R_0$ be the
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initial amount deposited.
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a. Find a recurrence relation for $R_0, R_1, R_2, \dots$. Justify your answer.
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b. If $R_0 = \$5,000$, find the am,ount of money on deposit at the end of one
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year.
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c. Find the APY for the account.
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38. _Compound Interest:_
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Suppose a certain amount of money is deposited in an account paying 3% annual
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interest compounded monthly. For each positive integer $n$, let
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$S_n = \text{ the amount on deposit at the end of the }$ $n$<sup>th</sup>
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month, and let $S_0$ be the initial amount deposited.
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a. Find a recurrence relation for $S_0, S_1, S_2, \dots$, assuming no additional
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deposits or withdrawals during the year. Justify your answer.
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b. If $S_0 = \$10,000$, find the amount of money on deposit at the end of one
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year.
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c. Find the APY for the account.
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39. With each step you take when climbing a staircase, you can move up either
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one stair or two stairs. As a result, you can climb the entire staircase
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taking one stair at a time, taking two at a time, or taking a combination of
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one-and two-stair increments. For each integer $n \geq 1$, if the staircase
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conssits of $n$ stairs, let $c_n$ be the number of different ways to climb
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the staircase. Find a recurrence relation for $c_1, c_2, c_3, \dots$.
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Justify your answer.
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40. A set of blocks contains blocks of heights $1$, $2$, and $4$ centimeters.
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Imagine constructing towers by piling blocks of different heights directly
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on top of one another. (A tower of height $6$ cm could be obtained using six
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$1$-cm blocks, three $2$-cm blocks one $2$-cm block with one $4$-cm block on
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top, one $4$-cm block with one $2$-cm block on top, and so forth.) Let $t_n$
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be the number of ways to construct a tower of height $n$ cm using blocks
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from the set. (Assume an unlimited supply of blocks of each size.) Find a
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recurrence relation for $t_1, t_2, t_3, \dots$. Justify your answer.
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41. Assume the truth of the distributive law (Appendix A, F3), and use the
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recursive definition of summation, together with mathematical induction, to
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prove the generalized distributive law that for every positive integer $n$,
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if $a_1, a_2, \dots, a_n$ and $c$ are real numbers, then
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$$ \sum_{i = 1}^{n}{ca_i} = c\left(\sum_{i = 1}^{n}{a_i}\right) $$
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42. Assume the truth of the commutative and associative laws (Appendix A, F1 and
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F2), and use the recursive definition of product, together with mathematical
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induction, to prove that for every positive integer $n$, if
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$a_1, a_2, \dots, a_n$ and $b_1, b_2, \dots, b_n$ are real numbers, then
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$$ \prod_{i = 1}^{n}{(a_ib_i)} = \left(\prod_{i = 1}^{n}{a_i}\right)\left(\prod_{i = 1}^{n}{b_i}\right) $$
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43. Assume the truth of the commutative and associative laws (Appendix A, F1 and
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F2), and use the recursive definition of product, together with mathematical
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induction, to prove that for each positive integer $n$, if
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$a_1, a_2, \dots, a_n$ and $c$ are real numbers, then
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$$ \prod_{i = 1}^{n}{(ca_i)} = c^n\left(\prod_{i = 1}^{n}{a_i}\right) $$
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44. The triangle inequality for absolute value states that for all real numbers
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$a$ and $b$, $|a + b| \leq |a| + |b|$. Use the recursive definition of
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summation, the triangle inequality, the definition of absolute value, and
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mathematical induction to prove that for each p ositive integer $n$, if
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$a_1, a_2, \dots, a_n$ are real numbers, then
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$$ \left| \sum_{i = 1}^{n}{a_i} \right| \leq \sum_{i = 1}^{n}{|a_i|} $$
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45. Prove that any sum of even integers is even.
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46. Prove that any sum of an odd number of odd integers is odd.
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47. Deduce from exercise 46 that for any positive integer $n$ if there is a sum
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of $n$ odd integers that is even, then $n$ is even.
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