Question 1

write the ﬁrst seven terms of the sequence determined by the generating function.
-$$( 1 + x ) ^ { 9 }$$

Accepted Answer

1, 9, 36,

Question 2

solve the recurrence relation either by using the characteristic equation or by discovering a pattern formed by the terms.
-$$a _ { n } = 5 a _ { n - 1 } - 4 a _ { n - 2 } , \quad a _ { 0 } = 0 , a _ { 1 } = 1$$

Accepted Answer

The answer of solve the recurrence relation either by using...

Question 3

describe each sequence recursively. Include initial conditions and assume that the sequences begin with aS1U1B11S1U1B0.
-an = the number of ways to go down an n-step staircase if you go down 1, 2, or 3 steps at a time

Accepted Answer

Question 4

Assume that the characteristic equation for a homogeneous linear recurrence relation with constant coeﬃ- cients is $$( r + 1 ) ^ { 4 } ( r - 1 ) ^ { 4 } = 0$$ Describe the form for the general solution to the recurrence relation.

Accepted Answer

The answer of Assume that the characteristic equation for a...

Question 5

If G(x) is the generating function for a0, a1, a2, a3, . . . , describe in terms of G(x) the generating function for 0, 0, 0, a0, a1, a2, . . . .

Accepted Answer

Question 6

Set up a generating function and use it to ﬁnd the number of ways in which eleven identical coins can be put in three distinct envelopes if each envelope has most six coins in it.

Accepted Answer

The answer of Set up a generating function and use...

Question 7

Set up a generating function and use it to ﬁnd the number of ways in which eleven identical coins can be put in three distinct envelopes (labeled A, B, C) if envelope A has at least three coins in it.

Accepted Answer

The answer of Set up a generating function and use...

Question 8

How many permutations of all 26 letters of the alphabet are there that contain at least one of the words DOG, BIG, OIL?

Accepted Answer

The answer of How many permutations of all 26 letters...

Question 9

determine whether the recurrence relation is a linear homogeneous recurrence relation with constant coeﬃcients.
-$$a _ { n } = 0.7 a _ { n - 1 } - 0.3 a _ { n - 2 }$$

Accepted Answer

The answer of determine whether the recurrence relation is a...

Question 10

ﬁnd the coeﬃcient of x8 in the power series of each of the function.
-$$\left( 1 + x ^ { 2 } + x ^ { 4 } + x ^ { 6 } \right) ^ { 3 }$$

Accepted Answer

The answer of ﬁnd the coeﬃcient of x8 in the...

Question 11

ﬁnd the coeﬃcient of x8 in the power series of each of the function.
-$$( 1 + x ) \left( 1 + x ^ { 2 } \right) \left( 1 + x ^ { 3 } \right) \left( 1 + x ^ { 4 } \right) \left( 1 + x ^ { 5 } \right)$$

Accepted Answer

The answer of ﬁnd the coeﬃcient of x8 in the...

Question 12

Use generating functions to solve $a _ { n } = 5 a _ { n - 1 } + 1 , a _ { 0 } = 1$

Accepted Answer

The answer of Use generating functions to solve \(a _...

Question 13

Find the number of positive integers $\leq 1000 \text { that are multiples of at least one of } 3,5,11 \text {. }$

Accepted Answer

The answer of Find the number of positive integers \(\leq...

Question 14

If G(x) is the generating function for a0, a1, a2, a3, . . . , describe in terms of G(x) the generating function for 0, 0, 0, a3, a4, a5, . . . .

Accepted Answer

Question 15

ﬁnd the coeﬃcient of x8 in the power series of each of the function.
-$$x ^ { 2 } / ( 1 + 2 x ) ^ { 2 }$$

Accepted Answer

The answer of ﬁnd the coeﬃcient of x8 in the...

Question 16

Set up a generating function and use it to ﬁnd the number of ways in which eleven identical coins can be put in three distinct envelopes (labeled A, B, C) envelopes A and B have the same number of coins in them.

Accepted Answer

The answer of Set up a generating function and use...

Question 17

describe each sequence recursively. Include initial conditions and assume that the sequences begin with aS1U1B11S1U1B0.
-an = the number of bit strings of length n with an even number of 0's

Accepted Answer

Question 18

If G(x) is the generating function for a0, a1, a2, a3, . . . , describe in terms of G(x) the generating function for a0, 0, a1, 0, a2, 0, a3, 0, a4, . . . .

Accepted Answer

Question 19

Assume that the characteristic equation for a homogeneous linear recurrence relation with constant coeﬃ- cients is $$( r - 3 ) ^ { 2 } ( r - 4 ) ^ { 3 } ( r + 7 ) ^ { 2 } = 0$$ Describe the form for the general solution to the recurrence relation.

Accepted Answer

The answer of Assume that the characteristic equation for a...

Question 20

If G(x) is the generating function for a0, a1, a2, a3, . . . , describe in terms of G(x) the generating function for 5, a1, 0, a3, a4, a5, . . . .

Accepted Answer

Exam 8: A: Advanced Counting Techniques

write the ﬁrst seven terms of the sequence determined by the generating function. - (1+x)9( 1 + x ) ^ { 9 }(1+x)9

solve the recurrence relation either by using the characteristic equation or by discovering a pattern formed by the terms. - an=5an−1−4an−2,a0=0,a1=1a _ { n } = 5 a _ { n - 1 } - 4 a _ { n - 2 } , \quad a _ { 0 } = 0 , a _ { 1 } = 1an​=5an−1​−4an−2​,a0​=0,a1​=1

describe each sequence recursively. Include initial conditions and assume that the sequences begin with a1. -an = the number of ways to go down an n-step staircase if you go down 1, 2, or 3 steps at a time

Assume that the characteristic equation for a homogeneous linear recurrence relation with constant coeﬃ- cients is (r+1)4(r−1)4=0( r + 1 ) ^ { 4 } ( r - 1 ) ^ { 4 } = 0(r+1)4(r−1)4=0 Describe the form for the general solution to the recurrence relation.

If G(x) is the generating function for a0, a1, a2, a3, . . . , describe in terms of G(x) the generating function for 0, 0, 0, a0, a1, a2, . . . .

Set up a generating function and use it to ﬁnd the number of ways in which eleven identical coins can be put in three distinct envelopes if each envelope has most six coins in it.

Set up a generating function and use it to ﬁnd the number of ways in which eleven identical coins can be put in three distinct envelopes (labeled A, B, C) if envelope A has at least three coins in it.

How many permutations of all 26 letters of the alphabet are there that contain at least one of the words DOG, BIG, OIL?

determine whether the recurrence relation is a linear homogeneous recurrence relation with constant coeﬃcients. - an=0.7an−1−0.3an−2a _ { n } = 0.7 a _ { n - 1 } - 0.3 a _ { n - 2 }an​=0.7an−1​−0.3an−2​

ﬁnd the coeﬃcient of x8 in the power series of each of the function. - (1+x2+x4+x6)3\left( 1 + x ^ { 2 } + x ^ { 4 } + x ^ { 6 } \right) ^ { 3 }(1+x2+x4+x6)3

ﬁnd the coeﬃcient of x8 in the power series of each of the function. - (1+x)(1+x2)(1+x3)(1+x4)(1+x5)( 1 + x ) \left( 1 + x ^ { 2 } \right) \left( 1 + x ^ { 3 } \right) \left( 1 + x ^ { 4 } \right) \left( 1 + x ^ { 5 } \right)(1+x)(1+x2)(1+x3)(1+x4)(1+x5)

Use generating functions to solve an=5an−1+1,a0=1a _ { n } = 5 a _ { n - 1 } + 1 , a _ { 0 } = 1an​=5an−1​+1,a0​=1

Find the number of positive integers ≤1000 that are multiples of at least one of 3,5,11. \leq 1000 \text { that are multiples of at least one of } 3,5,11 \text {. }≤1000 that are multiples of at least one of 3,5,11.

If G(x) is the generating function for a0, a1, a2, a3, . . . , describe in terms of G(x) the generating function for 0, 0, 0, a3, a4, a5, . . . .

ﬁnd the coeﬃcient of x8 in the power series of each of the function. - x2/(1+2x)2x ^ { 2 } / ( 1 + 2 x ) ^ { 2 }x2/(1+2x)2

Set up a generating function and use it to ﬁnd the number of ways in which eleven identical coins can be put in three distinct envelopes (labeled A, B, C) envelopes A and B have the same number of coins in them.

describe each sequence recursively. Include initial conditions and assume that the sequences begin with a1. -an = the number of bit strings of length n with an even number of 0's

If G(x) is the generating function for a0, a1, a2, a3, . . . , describe in terms of G(x) the generating function for a0, 0, a1, 0, a2, 0, a3, 0, a4, . . . .

Assume that the characteristic equation for a homogeneous linear recurrence relation with constant coeﬃ- cients is (r−3)2(r−4)3(r+7)2=0( r - 3 ) ^ { 2 } ( r - 4 ) ^ { 3 } ( r + 7 ) ^ { 2 } = 0(r−3)2(r−4)3(r+7)2=0 Describe the form for the general solution to the recurrence relation.

If G(x) is the generating function for a0, a1, a2, a3, . . . , describe in terms of G(x) the generating function for 5, a1, 0, a3, a4, a5, . . . .

The Foundations: Logic and Proofs

A: the Foundations: Logic and Proofs

Basic Structures: Sets, Functions, Sequences, Sums, Matrices

A: Basic Structures: Sets, Functions, Sequences, Sums, Matrices

Algorithms

A: Algorithms

Number Theory and Cryptography

A: Number Theory and Cryptography

Induction and Recursion

A: Induction and Recursion

Counting

A: Counting

Discrete Probability

A: Discrete Probability

Advanced Counting Techniques

Relations

A: Relations

Graphs

A: Graphs

Trees

A: Trees

Boolean Algebra

A: Boolean Algebra

Modeling Computation

A: Modeling Computation

Mathematics Problem Set: Set Theory, Number Theory, Combinatorics, and Boolean Algebra

Filters

write the ﬁrst seven terms of the sequence determined by the generating function. - $( 1 + x ) ^ { 9 }$

solve the recurrence relation either by using the characteristic equation or by discovering a pattern formed by the terms. - $a _ { n } = 5 a _ { n - 1 } - 4 a _ { n - 2 } , \quad a _ { 0 } = 0 , a _ { 1 } = 1$

describe each sequence recursively. Include initial conditions and assume that the sequences begin with a₁. -a_n = the number of ways to go down an n-step staircase if you go down 1, 2, or 3 steps at a time

Assume that the characteristic equation for a homogeneous linear recurrence relation with constant coeﬃ- cients is $( r + 1 ) ^ { 4 } ( r - 1 ) ^ { 4 } = 0$ Describe the form for the general solution to the recurrence relation.

If G(x) is the generating function for a₀, a₁, a₂, a₃, . . . , describe in terms of G(x) the generating function for 0, 0, 0, a₀, a₁, a₂, . . . .

determine whether the recurrence relation is a linear homogeneous recurrence relation with constant coeﬃcients. - $a _ { n } = 0.7 a _ { n - 1 } - 0.3 a _ { n - 2 }$

ﬁnd the coeﬃcient of x8 in the power series of each of the function. - $\left( 1 + x ^ { 2 } + x ^ { 4 } + x ^ { 6 } \right) ^ { 3 }$

ﬁnd the coeﬃcient of x8 in the power series of each of the function. - $( 1 + x ) \left( 1 + x ^ { 2 } \right) \left( 1 + x ^ { 3 } \right) \left( 1 + x ^ { 4 } \right) \left( 1 + x ^ { 5 } \right)$

Use generating functions to solve $a _ { n } = 5 a _ { n - 1 } + 1 , a _ { 0 } = 1$

Find the number of positive integers $\leq 1000 \text { that are multiples of at least one of } 3,5,11 \text {. }$

If G(x) is the generating function for a₀, a₁, a₂, a₃, . . . , describe in terms of G(x) the generating function for 0, 0, 0, a₃, a₄, a₅, . . . .

ﬁnd the coeﬃcient of x8 in the power series of each of the function. - $x ^ { 2 } / ( 1 + 2 x ) ^ { 2 }$

describe each sequence recursively. Include initial conditions and assume that the sequences begin with a₁. -a_n = the number of bit strings of length n with an even number of 0's

If G(x) is the generating function for a₀, a₁, a₂, a₃, . . . , describe in terms of G(x) the generating function for a0, 0, a₁, 0, a₂, 0, a₃, 0, a₄, . . . .

Assume that the characteristic equation for a homogeneous linear recurrence relation with constant coeﬃ- cients is $( r - 3 ) ^ { 2 } ( r - 4 ) ^ { 3 } ( r + 7 ) ^ { 2 } = 0$ Describe the form for the general solution to the recurrence relation.

If G(x) is the generating function for a₀, a₁, a₂, a₃, . . . , describe in terms of G(x) the generating function for 5, a₁, 0, a₃, a₄, a₅, . . . .