diff --git a/chapter_5/exercises.md b/chapter_5/exercises.md
index 3665be7..ca537d5 100644
--- a/chapter_5/exercises.md
+++ b/chapter_5/exercises.md
@@ -7141,3 +7141,332 @@ the Euclidean algorithm, find integers $u$ and $v$ so that
 $\text{gcd}(330, 156) = 330u + 156v$.
 
 Omitted.
+
+---
+
+**Exercise Set 5.6**
+
+Page 360
+
+Find the first four terms of each of the recursively defined sequences in 1-8.
+
+1. $a_k = 2a_{k - 1} + k$, for every integer $k \geq 2$ $a_1 = 1$
+
+2. $b_k = b_{k - 1} + 3_k$, for every integer $k \geq 2$ $b_1 = 1$
+
+3. $c_k = k(c_{k - 1})^2$, for every integer $k \geq 1$ $c_0 = 1$
+
+4. $d_k = k(d_{k - 1})^2$, for every integer $k \geq 1$ $d_0 = 3$
+
+5. $s_k = s_{k - 1} + 2s_{k - 2}$, for every integer $k \geq 2$, $s_0 = 1$,
+   $s_1 = 1$
+
+6. $t_k = t_{k - 1} + 2t_{k - 2}$, for every integer $k \geq 2$
+   $t_0 = -1, t_1 = 2$
+
+7. $u_k = ku_{k - 1} - u_{k - 2}$, for every integer $k \geq 3$
+   $u_1 = 1, u_2 = 1$
+
+8. $v_k = v_{k - 1} + v_{k - 2} + 1$, for every integer $k \geq 3$
+   $v_1 = 1, v_2 = 3$
+
+9. Let $a_0, a_1, a_2, \dots$ be defined by the formula $a_n = 3n + 1$, for
+   every integer $n \geq 0$. Show that this sequence satisfies the recurrence
+   relation $a_k = a_{k - 1} + 3$, for every integer $k \geq 1$.
+
+10. let $b_0, b_1, b_2, \dots$ be defined by the formula $b_n = 4^n$, for every
+    integer $n \geq 0$. Show that this sequence satisfies the recurrence
+    relation $b_k = 4b_{k - 1}$, for every integer $k \geq 1$.
+
+11. Let $c_0, c_1, c_2, \dots$ be defined by the formula $c_n = 2^n - 1$ for
+    every integer $n \geq 0$. Show that this sequence satisfies the recurrence
+    relation $c_k = 2c_{k  - 1} + 1$ for every integer $k \geq 1$.
+
+12. Let $s_0, s_1, s_2, \dots$ be defined by the formula
+    $s_n = \dfrac{(-1)^n}{n!}$ for every integer $n \geq 0$. Show that this
+    sequence satisfies the following recurrence relation for every integer
+    $k \geq 1$:
+
+$$ s_k = \frac{-s_{k - 1}}{k} $$
+
+13. Let $t_0, t_1, t_2, \dots$ be defined by the formula $t_n = 2 + n$ for every
+    integer $n \geq 0$. Show that this sequence satisfies the following
+    recurrence relation for every integer $k \geq 2$:
+
+$$ t_k = 2t_{k - 1} - t_{k - 2} $$
+
+14. Let $d_0, d_1, d_2, \dots$ be defined by the formula $d_n = 3^n - 2^n$ for
+    every integer $n \geq 0$. Show that this sequence satisfies the following
+    recurrence relation for every integer $k \geq 2$:
+
+$$ d_k = 5d_{k - 1} - 6d_{k - 2} $$
+
+15. For the sequence of Catalan numbers defined in Example 5.6.4, prove that for
+    each integer $n \geq 1$,
+
+$$ C_n = \frac{1}{4n + 2}\binom{2n + 2}{n + 1}$$
+
+16. Use the recurrence relation and values for the Tower of Hanoi sequence
+    $m_1, m_2, m_3, \dots$ discussed in Example 5.6.5 to compute $m_7$ and
+    $m_8$.
+
+17. _Tower of Hanoi with Adjacency Requirement:_
+
+Suppose that in addition to the requirement that they never move a larger disk
+on top of a smaller one, the priests who move the disks of the Tower of Hanoi
+are also allowed only to move disks one by one from one pole to an _adjacent_
+pole. Assume poles $A$ and $C$ are at the two ends of the row and pole $B$ is in
+the middle. Let
+
+$$ a_n = \left[\text{the minimum number of moves needed to transfer a tower of } n \text{ disks from pole } A \text{ to pole } C \right] $$
+
+a. Find $a_1, a_2$, and $a_3$.
+
+b. Find $a_4$.
+
+c. Find a recurrence relation for $a_1, a_2, a_3, \dots$. Justify your answer.
+
+18. _Tower of Hanoi with Adjacency Requirement:_
+
+Suppose the same situation as in exercise 17. Let
+
+$$ b_n = \left[\text{the minimum number of moves needed to transfer a tower of } n \text{ disks from pole } A \text{ to pole } B \right] $$
+
+a. Find $b_1, b_2$, and $b_3$.
+
+b. Find $b_4$.
+
+c. Show that $b_k = a_{k - 1} + 1 + b_{k - 1}$ for each integer $k \geq 2$,
+where $a_1, a_2, a_3, \dots$ is the sequence defined in exercise 17.
+
+d. Show that $b_k \leq 3b_{k - 1} + 1$ for each integer $k \geq 2$.
+
+e. Show that $b_k = 3b_{k - 1} + 1$ for each integer $k \geq 2$.
+
+19. _Four-Pole Tower of Hanoi:_
+
+Suppose that the Tower of Hanoi problem has four poles in a row instead of
+three. Disks can be transferred one by one from one pole to any other pole, but
+at no time may a larger disk be placed on top of a smaller disk. Let $s_n$ be
+the minimum number of moves needed to transfer the entire tower of $n$ disks
+from the left-most to the right-most pole.
+
+a. Find $s_1, s_2$, and $s_3$.
+
+b. Find $s_4$.
+
+c. Show that $s_k \leq 2s_{k - 2} + 3$ for every integer $k \geq 3$.
+
+20. _Tower of Hanoi Poles in a Circle:_
+
+Suppose that instead of being lined up in a row, the three poles for the
+original Tower of Hanoi are placed in a circle. The monks move the disks one by
+one from one pole to another, but they may only move disks one over in a
+clockwise direction and they may never move a larger disk on top of a smaller
+one. Let $c_n$ be the minimum number of moves needed to transfer a pile of $n$
+disks from one pole to the next adjacent pole in the clockwise direction.
+
+a. Justify the inequality $c_k \leq 4c_{k - 1} + 1$ for each integer $k \geq 2$.
+
+b. The expression $4c_{k - 1} + 1$ is not the minimum number of moves needed to
+transfer a pile of $k$ disks from one pole to another. Explain, for example, why
+$c_3 \neq 4c_2 + 1$.
+
+21. _Double Tower of Hanoi:_
+
+In this variation of the Tower of Hanoi there are three poles in a row and $2n$
+disks, two each of $n$ different sizes, where $n$ is any positive integer.
+Initially one of the poles contains all the disks placed on top of each other in
+pairs of decreasing size. Disks are transferred one by one from one pole to
+another, but at no time may a larger disk be placed on top of a smaller disk.
+However, a disk may be placed on top of one of the same size. Let $t_n$ be the
+minimum number of moves needed to transfer a tower of $2n$ disks from one pole
+to another.
+
+a. Find $t_1$ and $t_2$.
+
+b. Find $t_3$.
+
+c. Find a recurrence relation for $t_1, t_2, t_3, \dots$.
+
+22. _Fibonacci Variation:_
+
+A single pair of rabbits (male and female) is born at the beginning of a year.
+Assume the following conditions (which are somewhat more realistic than
+Fibonacci's):
+
+(1) Rabbit pairs are not fertile during their first months of life but
+thereafter give birth to four new male/female pairs at the end of every month.
+
+(2) No rabbits die.
+
+a. Let $r_n = \text{ the number of rabbits alive at the end of month } n$, for
+each integer $n \geq 1$, and let $r_0 = 1$. Find a recurrence relation for
+$r_0, r_1, r_2, \dots$. Justify your answer.
+
+b. Compute $r_0, r_1, r_2, r_3, r_4, r_5$, and $r_6$.
+
+c. How many rabbits will there be at the end of the year?
+
+23. _Fibonacci Variation:_
+
+A single pair of rabbits (male and female) is born at the beginning of a year.
+Assume the following conditions:
+
+(1) Rabbit pairs are not fertile during their first _two_ months of life but
+thereafter give birth to three new male/female pairs at the end of every month.
+
+(2) No rabbits die.
+
+a. Let
+$s_n = \text{ the number of pairs of rabbits alive at the end of month } n$, for
+each integer $n \geq 1$, and let $s_0 = 1$. Find a recurrence relation for
+$s_0, s_1, s_2, \dots$. Justify your answer.
+
+b. Compute $s_0, s_1, s_2, s_3, s_4$, and $s_5$.
+
+c. How many rabbits will there be at the end of the year?
+
+In 24-34, $F_0, F_1, F_2, \dots$ is the Fibonacci sequence.
+
+24. Use the recurrence relation and values for $F_0, F_1, F_2, \dots$ given in
+    Example 5.6.6 to compute $F_{13}$ and $F_{14}$.
+
+25. The Fibonacci sequence satisfies the recurrence relation
+    $F_k = F_{k - 1} + F_{k - 2}$, for every integer $k \geq 2$.
+
+a. Explain why the following is true:
+
+$$ F_{k + 1} = F_k + F_{k - 1} \text{ for each integer } k \geq 1 $$
+
+b. Write an equation expressing $F_{k + 2}$ in terms of $F_{k + 1}$ and $F_k$.
+
+c. Write an equation expressing $F_{k + 3}$ in terms of $F_{k + 2}$ and
+$F_{k + 1}$.
+
+26. Prove that $F_k = 3F_{k - 3} + 2F_{k - 4}$ for every integer $k \geq 4$.
+
+27. Prove that $F_k^2 - F_{k - 1}^2 = F_kF_{k + 1} - F_{k - 1}F_{k + 1}$, for
+    every integer $k \geq 1$.
+
+28. Prove that $F_{k + 1}^2 - F_k^2 - F_{k - 1}^2 = 2F_kF_{k - 1}$, for each
+    integer $k \geq 1$.
+
+29. Prove that $F_{k + 1}^2 - F_k^2 = F_{k - 1}F_{k + 2}$, for every integer
+    $k \geq 1$.
+
+30. Use mathematical induction to prove that for each integer $n \geq 0$,
+    $F_{n + 2}F_n - F_{n + 1}^2 = (-1)^n$.
+
+31. Use strong mathematical induction to prove that $F_n < 2^n$ for every
+    integer $n \geq 1$.
+
+32. Prove that for each integer $n \geq 0$, $\text{gcd}(F_{n + 1}, F_n) = 1$.
+    (The definition of $\text{gcd}$ is given in Section 4.10.)
+
+33. It turns out that the Fibonacci sequence satisfies the following explicit
+    formula: For every integer $F_n \geq 0$,
+
+$$ F_n = \frac{1}{\sqrt{5}}\left[\left(\frac{1 + \sqrt{5}}{2}\right)^{n + 1} - \left(\frac{1 - \sqrt{5}}{2}\right)^{n + 1}\right] $$
+
+Verify that the sequence defined by this formula satisfies the recurrence
+relation $F_k = F_{k - 1} + F_{k - 2}$ for every integer $k \geq 2$.
+
+34. (For students who have studied calculus) Find
+    $\lim\limits_{n \to \infty}\left(\dfrac{F_{n + 1}}{F_n}\right)$, assuming
+    that the limit exists.
+
+35. (For students who have studied calculus) Prove that
+    $\lim\limits_{n \to \infty}\left(\dfrac{F_{n + 1}}{F_n}\right)$ exists.
+
+36. (For students who have studied calculus) Define $x_0, x_1, x_2, \dots$ as
+    follows:
+
+$$ x_k = \sqrt{2 + x_{k - 1}} \quad \text{ for each integer } k \geq 1 $$
+
+$$ x_0 = 0 $$
+
+Find $\lim\limits_{n \to \infty}x_n$. (Assume that the limit exists.)
+
+37. _Compound Interest:_
+
+Suppose a certain amount of money is deposited in an account paying 4% annual
+interest compounded quarterly. For each positive integer $n$, let
+$R_n = \text{ the amount on deposit at the end of the }$ $n$<sup>th</sup>
+quarter, assuming no additional deposits or withdrawals, and let $R_0$ be the
+initial amount deposited.
+
+a. Find a recurrence relation for $R_0, R_1, R_2, \dots$. Justify your answer.
+
+b. If $R_0 = \$5,000$, find the am,ount of money on deposit at the end of one
+year.
+
+c. Find the APY for the account.
+
+38. _Compound Interest:_
+
+Suppose a certain amount of money is deposited in an account paying 3% annual
+interest compounded monthly. For each positive integer $n$, let
+$S_n  = \text{ the amount on deposit at the end of the }$ $n$<sup>th</sup>
+month, and let $S_0$ be the initial amount deposited.
+
+a. Find a recurrence relation for $S_0, S_1, S_2, \dots$, assuming no additional
+deposits or withdrawals during the year. Justify your answer.
+
+b. If $S_0 = \$10,000$, find the amount of money on deposit at the end of one
+year.
+
+c. Find the APY for the account.
+
+39. With each step you take when climbing a staircase, you can move up either
+    one stair or two stairs. As a result, you can climb the entire staircase
+    taking one stair at a time, taking two at a time, or taking a combination of
+    one-and two-stair increments. For each integer $n \geq 1$, if the staircase
+    conssits of $n$ stairs, let $c_n$ be the number of different ways to climb
+    the staircase. Find a recurrence relation for $c_1, c_2, c_3, \dots$.
+    Justify your answer.
+
+40. A set of blocks contains blocks of heights $1$, $2$, and $4$ centimeters.
+    Imagine constructing towers by piling blocks of different heights directly
+    on top of one another. (A tower of height $6$ cm could be obtained using six
+    $1$-cm blocks, three $2$-cm blocks one $2$-cm block with one $4$-cm block on
+    top, one $4$-cm block with one $2$-cm block on top, and so forth.) Let $t_n$
+    be the number of ways to construct a tower of height $n$ cm using blocks
+    from the set. (Assume an unlimited supply of blocks of each size.) Find a
+    recurrence relation for $t_1, t_2, t_3, \dots$. Justify your answer.
+
+41. Assume the truth of the distributive law (Appendix A, F3), and use the
+    recursive definition of summation, together with mathematical induction, to
+    prove the generalized distributive law that for every positive integer $n$,
+    if $a_1, a_2, \dots, a_n$ and $c$ are real numbers, then
+
+$$ \sum_{i = 1}^{n}{ca_i} = c\left(\sum_{i = 1}^{n}{a_i}\right) $$
+
+42. Assume the truth of the commutative and associative laws (Appendix A, F1 and
+    F2), and use the recursive definition of product, together with mathematical
+    induction, to prove that for every positive integer $n$, if
+    $a_1, a_2, \dots, a_n$ and $b_1, b_2, \dots, b_n$ are real numbers, then
+
+$$ \prod_{i = 1}^{n}{(a_ib_i)} = \left(\prod_{i = 1}^{n}{a_i}\right)\left(\prod_{i = 1}^{n}{b_i}\right) $$
+
+43. Assume the truth of the commutative and associative laws (Appendix A, F1 and
+    F2), and use the recursive definition of product, together with mathematical
+    induction, to prove that for each positive integer $n$, if
+    $a_1, a_2, \dots, a_n$ and $c$ are real numbers, then
+
+$$ \prod_{i = 1}^{n}{(ca_i)} = c^n\left(\prod_{i = 1}^{n}{a_i}\right) $$
+
+44. The triangle inequality for absolute value states that for all real numbers
+    $a$ and $b$, $|a + b| \leq |a| + |b|$. Use the recursive definition of
+    summation, the triangle inequality, the definition of absolute value, and
+    mathematical induction to prove that for each p ositive integer $n$, if
+    $a_1, a_2, \dots, a_n$ are real numbers, then
+
+$$ \left| \sum_{i = 1}^{n}{a_i} \right| \leq \sum_{i = 1}^{n}{|a_i|} $$
+
+45. Prove that any sum of even integers is even.
+
+46. Prove that any sum of an odd number of odd integers is odd.
+
+47. Deduce from exercise 46 that for any positive integer $n$ if there is a sum
+    of $n$ odd integers that is even, then $n$ is even.
diff --git a/chapter_5/notes.md b/chapter_5/notes.md
index 8310f3e..273437e 100644
--- a/chapter_5/notes.md
+++ b/chapter_5/notes.md
@@ -969,3 +969,34 @@ statement. Since statement IV is true (by assumption) and its hypothesis is true
 (by the argument just given), it follows (by modus ponens) that its conclusion
 is also true. That is, the values of all algorithm variables after execution of
 the loop are as specified in the post-condition for the loop.
+
+---
+
+Page 348
+
+**Definition**
+
+A **recurrence relation** for a sequence $a_0, a_1, a_2, \dots$ is a formula
+that relates each term $a_k$ to certain of its predecessors
+$a_{k - 1}, a_{k - 2}, \dots, a_{k - i}$, where $i$ is an integer with
+$k - i \geq 0$. If $i$ is a fixed integer, the **initial conditions** for such a
+recurrence relation specify the values of $a_0, a_1, a_2, \dots, a_{i - 1}$. If
+$i$ depends on $k$, the initial conditions specify the values of
+$a_0, a_1, \dots, a_m$, where $m$ is an integer with $m \geq 0$.
+
+---
+
+Page 358
+
+**Definition**
+
+Given numbers $a_1, a_2, \dots a_n$, where $n$ is a positive integer, the
+**summation from $i = 1$ to $n$ of the $a_i$**, denoted $\sum_{i = 1}^{n}{a_i}$,
+is defined as follows:
+
+$$ \sum_{i = 1}^{1}{a_i} = a_1 \quad \text{ and } \quad \sum_{i = 1}^{n}{a_i} = \left(\sum_{i = 1}^{n - 1}{a_i}\right) + a_n, \quad \text{ if } n > 1 $$
+
+The **product from $i = 1$ to $n$ of the $a_i$**, denoted
+$\prod_{i = 1}^{n}{a_i}$, is defined by
+
+$$ \prod_{i = 1}^{1}{a_i} = a_1 \quad \text{ and } \quad \prod_{i = 1}^{n}{a_i} = \left(\prod_{i = 1}^{n - 1}{a_i}\right) \cdot a_n, \quad \text{ if } \quad n > 1 $$
diff --git a/chapter_5/test_yourself.md b/chapter_5/test_yourself.md
index 0a7d5f9..2e2d137 100644
--- a/chapter_5/test_yourself.md
+++ b/chapter_5/test_yourself.md
@@ -143,3 +143,25 @@ the guard $G$ becomes false
 is true, then the values of the algorithm variables will be as specified _____.
 
 in the post-condition of the loop.
+
+---
+
+**Test Yourself**
+
+Page 359
+
+1. A recursive definition for a sequence consists of a _____ and _____.
+
+2. A recurrence relation is an equation that defines each later term of a
+   sequence by reference to _____ in the sequence.
+
+3. Initial conditions for a recursive definition of a sequence consist of one or
+   more of the _____ of the sequence.
+
+4. To solve a problem recursively means to divide the problem into smaller
+   subproblems of the same type as the initial problem, to suppose _____, and to
+   figure out how to use the supposition to _____.
+
+5. A crucial step for solving a problem recursively is to define a _____ in
+   terms of which the recurrence relation and initial conditions can be
+   specified.