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Unit 2: Recurrence Relations - Practice Quiz

MTH401 — Discrete Mathematics 60 Questions
0 Correct 0 Wrong 60 Left
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1 What is a recurrence relation?

Recurrence relation Easy
A. A formula for differentiating a function
B. A method for arranging a finite set
C. An equation defining terms using earlier terms
D. An equation containing only independent variables

2 Which of the following is a recurrence relation?

Recurrence relation Easy
A.
B.
C.
D.

3 Given and , what is ?

Recurrence relation Easy
A.
B.
C.
D.

4 A population begins with organisms and doubles every hour. Which recurrence relation models the population after hours?

Modelling with recurrence relations Easy
A. ,
B. ,
C. ,
D. ,

5 A person deposits dollars into an account at the end of every month, with no interest. If , which recurrence models the balance?

Modelling with recurrence relations Easy
A.
B.
C.
D.

6 A machine produces more items each day than it produced on the previous day. Which recurrence relation represents the daily production ?

Modelling with recurrence relations Easy
A.
B.
C.
D.

7 Which recurrence relation is homogeneous and linear with constant coefficients?

Homogeneous linear recurrence relations with constant coefficients Easy
A.
B.
C.
D.

8 What is the characteristic equation of ?

Homogeneous linear recurrence relations with constant coefficients Easy
A.
B.
C.
D.

9 What are the characteristic roots of ?

Homogeneous linear recurrence relations with constant coefficients Easy
A. and
B. and
C. and
D. and

10 If a second-order homogeneous recurrence has distinct characteristic roots and , what is the general form of its solution?

Homogeneous linear recurrence relations with constant coefficients Easy
A.
B.
C.
D.

11 For the shift operator , which expression correctly defines its action on ?

Method of inverse operator to solve the non-homogeneous recurrence relation with constant coefficient Easy
A.
B.
C.
D.

12 If a non-homogeneous recurrence is written as , what is its formal particular solution using the inverse operator?

Method of inverse operator to solve the non-homogeneous recurrence relation with constant coefficient Easy
A.
B.
C.
D.

13 For , which constant is a particular solution?

Method of inverse operator to solve the non-homogeneous recurrence relation with constant coefficient Easy
A.
B.
C.
D.

14 What is the ordinary generating function of a sequence ?

Generating functions Easy
A.
B.
C.
D.

15 What is the generating function of the sequence ?

Generating functions Easy
A.
B.
C.
D.

16 In the generating function , what is the coefficient of ?

Generating functions Easy
A.
B.
C.
D.

17 Which sequence has the generating function ?

Generating functions Easy
A.
B.
C.
D.

18 For with , what is the generating function ?

Solution of recurrence relation using generating functions Easy
A.
B.
C.
D.

19 For with , what is the generating function ?

Solution of recurrence relation using generating functions Easy
A.
B.
C.
D.

20 For the Fibonacci recurrence with and , what is its generating function?

Solution of recurrence relation using generating functions Easy
A.
B.
C.
D.

21 What are the order and degree of the recurrence relation ?

Recurrence relation Medium
A. Order and degree
B. Order and degree
C. Order and degree
D. Order and degree

22 Which sequence satisfies for , with and ?

Recurrence relation Medium
A.
B.
C.
D.

23 An account earns interest each month, after which is deposited. If is the balance after the deposit in month , which recurrence models the account?

Modelling with recurrence relations Medium
A.
B.
C.
D.

24 Let be the number of binary strings of length that contain no two consecutive s. Which recurrence and initial values correctly model ?

Modelling with recurrence relations Medium
A. , ,
B. , ,
C. , ,
D. , ,

25 What is the general solution of ?

Homogeneous linear recurrence relations with constant coefficients Medium
A.
B.
C.
D.

26 Which expression is the general solution of ?

Homogeneous linear recurrence relations with constant coefficients Medium
A.
B.
C.
D.

27 Solve given and .

Homogeneous linear recurrence relations with constant coefficients Medium
A.
B.
C.
D.

28 A second-order homogeneous recurrence has characteristic roots and . Which recurrence corresponds to these roots?

Homogeneous linear recurrence relations with constant coefficients Medium
A.
B.
C.
D.

29 For the recurrence , which expression is a valid particular solution?

Method of inverse operator to solve the non-homogeneous recurrence relation with constant coefficient Medium
A.
B.
C.
D.

30 For , which trial form accounts for resonance with the complementary solution?

Method of inverse operator to solve the non-homogeneous recurrence relation with constant coefficient Medium
A.
B.
C.
D.

31 Which expression is a particular solution of ?

Method of inverse operator to solve the non-homogeneous recurrence relation with constant coefficient Medium
A.
B.
C.
D.

32 Solve for , given .

Method of inverse operator to solve the non-homogeneous recurrence relation with constant coefficient Medium
A.
B.
C.
D.

33 What is the ordinary generating function of the sequence ?

Generating functions Medium
A.
B.
C.
D.

34 If , which expression equals ?

Generating functions Medium
A.
B.
C.
D.

35 If and , what is ?

Generating functions Medium
A.
B.
C.
D.

36 The generating function of a sequence is . Which formula describes its coefficients?

Generating functions Medium
A. and for
B. and for
C. and for
D. and for

37 Let for , with and . What is its generating function ?

Solution of recurrence relation using generating functions Medium
A.
B.
C.
D.

38 For with , which generating function represents the sequence?

Solution of recurrence relation using generating functions Medium
A.
B.
C.
D.

39 If , what is the coefficient in ?

Solution of recurrence relation using generating functions Medium
A.
B.
C.
D.

40 For the recurrence with , which equation for the generating function is correct?

Solution of recurrence relation using generating functions Medium
A.
B.
C.
D.

41 Consider the recurrence with . Which statement correctly describes its solutions?

Recurrence relation Hard
A. Every term from onward can be selected independently because the coefficient vanishes at .
B. No sequence satisfies the recurrence.
C. Exactly one sequence satisfies the recurrence.
D. Infinitely many sequences satisfy it, one for each choice of .

42 Let be the number of circular binary strings of length , with labeled positions, in which no two adjacent positions contain ; the first and last positions are also adjacent. Which recurrence and initial values are correct?

Modelling with recurrence relations Hard
A. for , with and
B. for , with and
C. for , with and
D. for , with and

43 Let denote the number of permutations of objects having no fixed points. Which recurrence correctly models ?

Modelling with recurrence relations Hard
A. , with and
B. , with and
C. , because the image of the first object can be chosen in ways before arranging all remaining objects.
D. , with and

44 Solve given , , and .

Homogeneous linear recurrence relations with constant coefficients Hard
A.
B.
C.
D.

45 Find the solution of with and .

Homogeneous linear recurrence relations with constant coefficients Hard
A.
B.
C.
D.

46 A sequence satisfies the recurrence whose characteristic polynomial is , with initial values . Under which condition does exist and remain nonzero?

Homogeneous linear recurrence relations with constant coefficients Hard
A. and
B. and
C. and
D. and

47 What is the minimum-order homogeneous linear recurrence with constant coefficients that can have as a solution?

Homogeneous linear recurrence relations with constant coefficients Hard
A. An order- recurrence with characteristic polynomial
B. An order- recurrence with characteristic polynomial
C. An order- recurrence with characteristic polynomial
D. An order- recurrence with characteristic polynomial

48 Using the shift operator , find a particular solution of equivalently .

Method of inverse operator to solve the non-homogeneous recurrence relation with constant coefficient Hard
A.
B.
C.
D.

49 Which expression is a particular solution of

Method of inverse operator to solve the non-homogeneous recurrence relation with constant coefficient Hard
A.
B.
C.
D.

50 Find a particular solution of

Method of inverse operator to solve the non-homogeneous recurrence relation with constant coefficient Hard
A.
B.
C.
D.

51 For which trial form contains the necessary terms for a particular solution without duplicating homogeneous terms?

Method of inverse operator to solve the non-homogeneous recurrence relation with constant coefficient Hard
A.
B. , since both forcing components must always be multiplied by the full resonance factor.
C.
D.

52 Let , , and . What is

Generating functions Hard
A.
B.
C.
D.

53 Determine the coefficient of in

Generating functions Hard
A.
B.
C.
D.

54 Suppose Which initial values and recurrence are implied?

Generating functions Hard
A. , , , and for
B. , , , and for
C. , , , and for
D. , , , and for

55 Let when and otherwise. What is the ordinary generating function ?

Generating functions Hard
A.
B.
C.
D.

56 For the Fibonacci sequence , , define Which expression equals ?

Generating functions Hard
A.
B.
C.
D.

57 Solve with .

Solution of recurrence relation using generating functions Hard
A.
B.
C.
D.

58 Solve with and .

Solution of recurrence relation using generating functions Hard
A.
B.
C.
D.

59 Let and If , which formal power series is ?

Solution of recurrence relation using generating functions Hard
A.
B.
C.
D.

60 A sequence is defined by and Which closed form is correct?

Solution of recurrence relation using generating functions Hard
A. and for
B. and for
C. and for
D. and for
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