discrete_mathematics_with_applications/chapter_5/test_yourself.md

Test Yourself

Page 296

1. The notation \sum_{k = m}^{n}{a_k} is read "_____."

The summation from k equals m to n of a sub k.

2. The expanded form of \sum_{k = m}^{n}{a_k} is _____.

 a_m + a_{m + 1} + a_{m + 2} + \dots + a_n 

3. The value of a_1 + a_2 + a_3 + \dots + a_n when n = 2 is "_____."

 a_1 + a_2 

4. The notation \prod_{k = m}^{n}{a_k} is read "_____."

The product from k equals m to n of a sub k.

5. If n is a positive integer, then n! = _____.

 n \cdot (n - 1) \dots \cdot 3 \cdot 2 \cdot 1 

6. \sum_{k = m}^{n}{a_k} + c\sum_{k = m}^{n}{b_k} = _____.

 \sum_{k = m}^{n}{a_k + cb_k} 

7. \left(\prod_{k = m}^{n}{a_k}\right)\left(\prod_{k = m}^{n}{b_k}\right) = _____.

 \prod_{k = m}^{n}{a_kb_k} 

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Test Yourself

Page 309

1. Mathematical induction is a method for proving that a property defined for integers n is true for all values of n that are _____.

greater than or equal to some initial value.

2. Let P(n) be a property defined for integers n and consider constructing a proof by mathematical induction for the statement "P(n) is true for all n \geq a."

a. In the basis step one must show _____.

that P(a) is true.

b. In the inductive step one supposes that _____ for a particular but arbitrarily chosen value of an integer k \geq a. This supposition is called the _____. One then has to show that _____.

P(k) is true; inductive hypothesis; P(k + 1) is true.

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Test Yourself

Page 320

1. Mathematical induction differs from the kind of induction used in the natural sciences because it is actually a form of _____ reasoning.

deductive

2. Mathematical induction can be used to _____ conjectures that have been made using inductive reasoning.

prove

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Test Yourself

Page 333

1. In a proof by strong mathematical induction the basis step may require checking a property P(n) for more _____ value of n.

than one

2. Suppose that in the basis step for a proof by strong mathematical induction the property P(n) was checked for every integer n from a through b. Then in the inductive step one assumes that for any integer k \geq b, the property P(n) is true for all values of i from _____ through _____ and one shows that _____ is true.

a; k; P(k + 1)

3. According to the well-ordering principle for the integers, if a set S of integers contains at least _____ and if there is some integer that is less than or equal to every _____, then _____.

one integer; integer in S; S contains a least element.

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Test Yourself

Page 346

1. A pre-condition for an algorithm is _____ and a post-condition for an algorithm is _____.

a predicate that describes the initial state of the input variables of the algorithm; a predicate that describes the final state of the output variables for the algorithm

2. A loop is defined as correct with respect to its pre- and post-conditions if, and only if, whenever the algorithm variables satisfy the pre-condition for the loop and the loop terminates after a finite number of steps, then _____.

the algorithm variables satisfy the post-condition for the loop

3. For each iteration of a loop, if a loop invariant is true before iteration of the loop, then _____.

it is true after iteration of the loop

4. Given a while loop with guard G and a predicate I(n) if the following four properties are true, then the loop is correct with respect to its pre- and post-conditions:

(a) The pre-condition for the loop implies that _____ before the first iteration of the loop.

I(0) is true

(b) For every integer k \geq 0, if the guard G and the predicate I(k) are both true before an iteration of the loop, then _____.

I(k + 1) is true after the iteration of the loop

c After a finite number of iterations of the loop, _____.

the guard G becomes false

(d) If N is the least number of iterations after which G is false and I(N) is true, then the values of the algorithm variables will be as specified _____.

in the post-condition of the loop.

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Test Yourself

Page 359

1. A recursive definition for a sequence consists of a _____ and _____.

recurrence relation; initial conditions

2. A recurrence relation is an equation that defines each later term of a sequence by reference to _____ in the sequence.

earlier terms

3. Initial conditions for a recursive definition of a sequence consist of one or more of the _____ of the sequence.

values of the first few terms

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 _____.

that the smaller subproblems have already been solved; solve the initial problem

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.

sequence

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Page 372

Test Yourself

1. To use iteration to find an explicit formula for a recursively defined sequence, start with the _____ and use successive substitution into the _____ to look for a numerical pattern.

initial conditions; recurrence relation

2. At every step of the iteration process, it is important to eliminate _____.

parentheses

3. If a single number, say a, is added to itself k times in one of the steps of the iteration, replace the sum by the expression _____.

k \cdot a

4. If a single number, say a, is multiplied by itself k times in one of the steps of the iteration, replace the product by the expression _____.

a^k

5. A general arithmetic sequence a_0, a_1, a_2, \dots with initial value a_0 and fixed constant summand d satisfies the recurrence relation _____ and has the explicit formula _____.

a_k = a_{k - 1} + d; a_n = a_0 + dn

6. A general geometric sequence a_0, a_1, a_2, \dots with initial value a_0 and fixed constant multiplier r satisfies the recurrence relation _____ and has the explicit formula _____.

a_k = ra_{k - 1}; a_n = r^na_0

7. When an explicit formula for a recursively defined sequence has been obtained by iteration, its correctness can be checked by _____.

mathematical induction

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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 _____.