Exam 8: A: Advanced Counting Techniques

Assume that the characteristic equation for a homogeneous linear recurrence relation with constant coeﬃ- cients is $( r - 5 ) ^ { 3 } = 0$ Describe the form for the general solution to the recurrence relation.

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(Short Answer)

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Question 1

Correct Answer:

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$a _ { n } = c 5 ^ { n } + d n 5 ^ { n } + e n ^ { 2 } 5 ^ { n }$

Suppose $f ( n ) = 4 f ( n / 2 ) + n + 2 , f ( 1 ) = 2$ . Find $f ( 8 )$ .

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Question 2

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226

solve the recurrence relation either by using the characteristic equation or by discovering a pattern formed by the terms. - $a _ { n } = a _ { n - 1 } + 3 n , \quad a _ { 0 } = 5$

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Question 3

Correct Answer:

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$a _ { n } = 5 + 3 \frac { n ( n + 1 ) } { 2 }$

write the ﬁrst seven terms of the sequence determined by the generating function. - $1 / ( 1 - 3 x )$

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Question 4

ﬁnd a closed form for the generating function for the sequence. -0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1 . . .

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Question 5

What form does a particular solution of the linear nonhomogeneous recurrence relation $a _ { n } = 4 a _ { n - 1 } - 4 a _ { n - 2 }$ + $F ( n )$ have when $F ( n ) = n ^ { 2 } \cdot 4 ^ { n }$

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Question 6

ﬁnd a closed form for the generating function for the sequence. -2, 4, 6, 8, 10, 12, . . .

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Question 7

solve the recurrence relation either by using the characteristic equation or by discovering a pattern formed by the terms. - $a _ { n } = 3 n a _ { n - 1 } , \quad a _ { 0 } = 2$

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Question 8

Suppose $|A|$ =8 and $| B |$ =4 . Find the number of functions $f : A \rightarrow B$ that are onto $B$ .

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Question 9

Set up a generating function and use it to ﬁnd the number of ways in which nine identical blocks can be given to four children, if the oldest child gets three blocks.

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Question 10

What form does a particular solution of the linear nonhomogeneous recurrence relation $a _ { n } = 4 a _ { n - 1 } - 4 a _ { n - 2 }$ + $F ( n )$ have when $F ( n ) = 2 ^ { n }$ ?

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Question 11

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 no envelope is empty.

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Question 12

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 that contain a pair of consecutive 0's

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Question 13

How many permutations of all 26 letters of the alphabet are there that contain none of the words: SAVE, PLAY, SNOW?

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Question 14

Consider the recurrence relation $a _ { n } = 2 a _ { n - 1 } + 3 n$ (a) Write the associated homogeneous recurrence relation. (b) Find the general solution to the associated homogeneous recurrence relation. (c) Find a particular solution to the given recurrence relation. (d) Write the general solution to the given recurrence relation.

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Question 15

Use generating functions to solve $a _ { n } = 5 a _ { n - 1 } + 3 , a _ { 0 } = 2$

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Question 16

determine whether the recurrence relation is a linear homogeneous recurrence relation with constant coeﬃcients. - $a _ { n } = a _ { n - 3 }$

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Question 17

write the ﬁrst seven terms of the sequence determined by the generating function. -cos x

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Question 18

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

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Question 19

ﬁnd the coeﬃcient of x8 in the power series of each of the function. - $\left( 1 + x ^ { 3 } \right) ^ { 12 }$

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Question 20

Exam 8: A: Advanced Counting Techniques

Assume that the characteristic equation for a homogeneous linear recurrence relation with constant coeﬃ- cients is (r−5)3=0( r - 5 ) ^ { 3 } = 0(r−5)3=0 Describe the form for the general solution to the recurrence relation.

Suppose f(n)=4f(n/2)+n+2,f(1)=2f ( n ) = 4 f ( n / 2 ) + n + 2 , f ( 1 ) = 2f(n)=4f(n/2)+n+2,f(1)=2 . Find f(8)f ( 8 )f(8) .

solve the recurrence relation either by using the characteristic equation or by discovering a pattern formed by the terms. - an=an−1+3n,a0=5a _ { n } = a _ { n - 1 } + 3 n , \quad a _ { 0 } = 5an​=an−1​+3n,a0​=5

write the ﬁrst seven terms of the sequence determined by the generating function. - 1/(1−3x)1 / ( 1 - 3 x )1/(1−3x)

ﬁnd a closed form for the generating function for the sequence. -0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1 . . .

What form does a particular solution of the linear nonhomogeneous recurrence relation an=4an−1−4an−2a _ { n } = 4 a _ { n - 1 } - 4 a _ { n - 2 }an​=4an−1​−4an−2​ + F(n)F ( n )F(n) have when F(n)=n2⋅4nF ( n ) = n ^ { 2 } \cdot 4 ^ { n }F(n)=n2⋅4n

ﬁnd a closed form for the generating function for the sequence. -2, 4, 6, 8, 10, 12, . . .

solve the recurrence relation either by using the characteristic equation or by discovering a pattern formed by the terms. - an=3nan−1,a0=2a _ { n } = 3 n a _ { n - 1 } , \quad a _ { 0 } = 2an​=3nan−1​,a0​=2

Suppose ∣A∣|A|∣A∣ =8 and ∣B∣| B |∣B∣ =4 . Find the number of functions f:A→Bf : A \rightarrow Bf:A→B that are onto BBB .

Set up a generating function and use it to ﬁnd the number of ways in which nine identical blocks can be given to four children, if the oldest child gets three blocks.

What form does a particular solution of the linear nonhomogeneous recurrence relation an=4an−1−4an−2a _ { n } = 4 a _ { n - 1 } - 4 a _ { n - 2 }an​=4an−1​−4an−2​ + F(n)F ( n )F(n) have when F(n)=2nF ( n ) = 2 ^ { n }F(n)=2n ?

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 no envelope is empty.

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

How many permutations of all 26 letters of the alphabet are there that contain none of the words: SAVE, PLAY, SNOW?

Use generating functions to solve an=5an−1+3,a0=2a _ { n } = 5 a _ { n - 1 } + 3 , a _ { 0 } = 2an​=5an−1​+3,a0​=2

determine whether the recurrence relation is a linear homogeneous recurrence relation with constant coeﬃcients. - an=an−3a _ { n } = a _ { n - 3 }an​=an−3​

write the ﬁrst seven terms of the sequence determined by the generating function. -cos x

ﬁnd the coeﬃcient of x8 in the power series of each of the function. - 1/(1−3x2)1 / \left( 1 - 3 x ^ { 2 } \right)1/(1−3x2)

ﬁnd the coeﬃcient of x8 in the power series of each of the function. - (1+x3)12\left( 1 + x ^ { 3 } \right) ^ { 12 }(1+x3)12

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

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Assume that the characteristic equation for a homogeneous linear recurrence relation with constant coeﬃ- cients is $( r - 5 ) ^ { 3 } = 0$ Describe the form for the general solution to the recurrence relation.

Suppose $f ( n ) = 4 f ( n / 2 ) + n + 2 , f ( 1 ) = 2$ . Find $f ( 8 )$ .

solve the recurrence relation either by using the characteristic equation or by discovering a pattern formed by the terms. - $a _ { n } = a _ { n - 1 } + 3 n , \quad a _ { 0 } = 5$

write the ﬁrst seven terms of the sequence determined by the generating function. - $1 / ( 1 - 3 x )$

What form does a particular solution of the linear nonhomogeneous recurrence relation $a _ { n } = 4 a _ { n - 1 } - 4 a _ { n - 2 }$ + $F ( n )$ have when $F ( n ) = n ^ { 2 } \cdot 4 ^ { n }$

solve the recurrence relation either by using the characteristic equation or by discovering a pattern formed by the terms. - $a _ { n } = 3 n a _ { n - 1 } , \quad a _ { 0 } = 2$

Suppose $|A|$ =8 and $| B |$ =4 . Find the number of functions $f : A \rightarrow B$ that are onto $B$ .

What form does a particular solution of the linear nonhomogeneous recurrence relation $a _ { n } = 4 a _ { n - 1 } - 4 a _ { n - 2 }$ + $F ( n )$ have when $F ( n ) = 2 ^ { n }$ ?

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 that contain a pair of consecutive 0's

Use generating functions to solve $a _ { n } = 5 a _ { n - 1 } + 3 , a _ { 0 } = 2$

determine whether the recurrence relation is a linear homogeneous recurrence relation with constant coeﬃcients. - $a _ { n } = a _ { n - 3 }$

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

ﬁnd the coeﬃcient of x8 in the power series of each of the function. - $\left( 1 + x ^ { 3 } \right) ^ { 12 }$