292 lines
8.6 KiB
Markdown
292 lines
8.6 KiB
Markdown
**Test Yourself**
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Page 296
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1. The notation $\sum_{k = m}^{n}{a_k}$ is read "_____."
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The summation from $k$ equals $m$ to $n$ of $a$ sub $k$.
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2. The expanded form of $\sum_{k = m}^{n}{a_k}$ is _____.
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$$ a_m + a_{m + 1} + a_{m + 2} + \dots + a_n $$
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3. The value of $a_1 + a_2 + a_3 + \dots + a_n$ when $n = 2$ is "_____."
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$$ a_1 + a_2 $$
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4. The notation $\prod_{k = m}^{n}{a_k}$ is read "_____."
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The product from $k$ equals $m$ to $n$ of $a$ sub $k$.
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5. If $n$ is a positive integer, then $n! =$ _____.
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$$ n \cdot (n - 1) \dots \cdot 3 \cdot 2 \cdot 1 $$
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6. $\sum_{k = m}^{n}{a_k} + c\sum_{k = m}^{n}{b_k} =$ _____.
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$$ \sum_{k = m}^{n}{a_k + cb_k} $$
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7. $\left(\prod_{k = m}^{n}{a_k}\right)\left(\prod_{k = m}^{n}{b_k}\right) =$
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_____.
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$$ \prod_{k = m}^{n}{a_kb_k} $$
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---
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**Test Yourself**
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Page 309
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1. Mathematical induction is a method for proving that a property defined for
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integers $n$ is true for all values of $n$ that are _____.
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greater than or equal to some initial value.
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2. Let $P(n)$ be a property defined for integers $n$ and consider constructing a
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proof by mathematical induction for the statement "P(n) is true for all
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$n \geq a$."
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a. In the basis step one must show _____.
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that $P(a)$ is true.
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b. In the inductive step one supposes that _____ for a particular but
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arbitrarily chosen value of an integer $k \geq a$. This supposition is called
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the _____. One then has to show that _____.
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$P(k)$ is true; inductive hypothesis; $P(k + 1)$ is true.
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---
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**Test Yourself**
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Page 320
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1. Mathematical induction differs from the kind of induction used in the natural
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sciences because it is actually a form of _____ reasoning.
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deductive
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2. Mathematical induction can be used to _____ conjectures that have been made
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using inductive reasoning.
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prove
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---
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**Test Yourself**
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Page 333
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1. In a proof by strong mathematical induction the basis step may require
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checking a property $P(n)$ for more _____ value of $n$.
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than one
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2. Suppose that in the basis step for a proof by strong mathematical induction
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the property $P(n)$ was checked for every integer $n$ from $a$ through $b$.
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Then in the inductive step one assumes that for any integer $k \geq b$, the
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property $P(n)$ is true for all values of $i$ from _____ through _____ and
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one shows that _____ is true.
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$a$; $k$; $P(k + 1)$
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3. According to the well-ordering principle for the integers, if a set $S$ of
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integers contains at least _____ and if there is some integer that is less
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than or equal to every _____, then _____.
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one integer; integer in $S$; $S$ contains a least element.
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---
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**Test Yourself**
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Page 346
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1. A pre-condition for an algorithm is _____ and a post-condition for an
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algorithm is _____.
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a predicate that describes the initial state of the input variables of the
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algorithm; a predicate that describes the final state of the output variables
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for the algorithm
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2. A loop is defined as correct with respect to its pre- and post-conditions if,
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and only if, whenever the algorithm variables satisfy the pre-condition for
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the loop and the loop terminates after a finite number of steps, then _____.
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the algorithm variables satisfy the post-condition for the loop
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3. For each iteration of a loop, if a loop invariant is true before iteration of
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the loop, then _____.
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it is true after iteration of the loop
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4. Given a **while** loop with guard $G$ and a predicate $I(n)$ if the following
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four properties are true, then the loop is correct with respect to its pre-
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and post-conditions:
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(a) The pre-condition for the loop implies that _____ before the first iteration
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of the loop.
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$I(0)$ is true
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(b) For every integer $k \geq 0$, if the guard $G$ and the predicate $I(k)$ are
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both true before an iteration of the loop, then _____.
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$I(k + 1)$ is true after the iteration of the loop
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\(c\) After a finite number of iterations of the loop, _____.
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the guard $G$ becomes false
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(d) If $N$ is the least number of iterations after which $G$ is false and $I(N)$
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is true, then the values of the algorithm variables will be as specified _____.
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in the post-condition of the loop.
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---
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**Test Yourself**
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Page 359
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1. A recursive definition for a sequence consists of a _____ and _____.
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recurrence relation; initial conditions
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2. A recurrence relation is an equation that defines each later term of a
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sequence by reference to _____ in the sequence.
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earlier terms
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3. Initial conditions for a recursive definition of a sequence consist of one or
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more of the _____ of the sequence.
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values of the first few terms
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4. To solve a problem recursively means to divide the problem into smaller
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subproblems of the same type as the initial problem, to suppose _____, and to
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figure out how to use the supposition to _____.
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that the smaller subproblems have already been solved; solve the initial problem
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5. A crucial step for solving a problem recursively is to define a _____ in
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terms of which the recurrence relation and initial conditions can be
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specified.
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sequence
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---
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Page 372
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**Test Yourself**
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1. To use iteration to find an explicit formula for a recursively defined
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sequence, start with the _____ and use successive substitution into the _____
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to look for a numerical pattern.
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initial conditions; recurrence relation
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2. At every step of the iteration process, it is important to eliminate _____.
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parentheses
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3. If a single number, say $a$, is added to itself $k$ times in one of the steps
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of the iteration, replace the sum by the expression _____.
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$k \cdot a$
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4. If a single number, say $a$, is multiplied by itself $k$ times in one of the
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steps of the iteration, replace the product by the expression _____.
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$a^k$
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5. A general arithmetic sequence $a_0, a_1, a_2, \dots$ with initial value $a_0$
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and fixed constant summand $d$ satisfies the recurrence relation _____ and
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has the explicit formula _____.
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$a_k = a_{k - 1} + d$; $a_n = a_0 + dn$
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6. A general geometric sequence $a_0, a_1, a_2, \dots$ with initial value $a_0$
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and fixed constant multiplier $r$ satisfies the recurrence relation _____ and
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has the explicit formula _____.
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$a_k = ra_{k - 1}$; $a_n = r^na_0$
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7. When an explicit formula for a recursively defined sequence has been obtained
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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 _____.
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$a_k = Aa_{k - 1} + Ba_{k - }$; 2; $A$ and $B$ are fixed real numbers with
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$B \neq 0$.
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2. Given a recurrence relation of the form $a_k = Aa_{k - 1} + Ba_{k - 2}$ for
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every integer $k \geq 2$, the characteristic equation of the relation is
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_____.
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$t^2 - At - B = 0$
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3. If a sequence $a_1, a_2, a_3, \dots$ is defined by a second-order linear
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homogeneous recurrence relation with constant coefficients and the
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characteristic equation for the relation has two distinct roots $r$ and $s$
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(which could be complex numbers), then the sequence is given by an explicit
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formula of the form _____.
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$a_n = Cr_n + Ds_n$ for every integer $n \geq 0$ where $C$ and $D$ are real or
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complex numbers.
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4. If a sequence $a_1, a_2, a_3, \dots$ is defined by a second-order linear
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homogeneous recurrence relation with constant coefficients and the
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characteristic equation for the relation has only a single root $r$, then the
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sequence is given by an explicit formula of the form _____.
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$a_n = Cr^n + Dnr^n$ where $C$ and $D$ are real numbers.
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---
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Page 397
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**Test Yourself**
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1. The base for a recursive definition of a set is _____.
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a statement that certain objects belong to the set
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2. The recursion for a recursive definition of a set is _____.
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a collection of rules indicating how to form new set objects from those already
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known to be in the set
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3. The restriction for a recursive definition of a set is _____.
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a statement that no objects belong to the set other than those coming from the
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base and the recursion
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4. One way to show that a given element is in a recursively defined set is to
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start with an element or elements in the _____ and apply the rules from the
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_____ until you obtain the given element.
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base; recursion
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5. To use structural induction to prove that every element in a recursively
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defined set $S$ satisfies a certain property, you show that _____ and that,
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for each rule in the recursion, if _____ then _____.
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each object in the base satisfies the property; the rule is applied to the
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objects in the base; the objects defined by the rule also satisfy the property
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6. A function is said to be defined recursively if, and only if, _____.
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its rule of definition refers to itself
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