Exam 8: Advanced Counting Techniques

Find $\mid A _ { 1 } \cup A _ { 2 } \cup A _ { 3 } \cup A _ { 4 }|$ if each set A_i has 100 elements, each intersection of two sets has 60 elements, each intersection of three sets has 20 elements, and there are 10 elements in all four sets.

(Short Answer)

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Question 121

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

(Short Answer)

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Question 122

In questions ﬁnd a closed form for the generating function for the sequence. - $\left( \begin{array} { c } 50 \\1\end{array} \right) , 2 \left( \begin{array} { c } 50 \\2\end{array} \right) , 3 \left( \begin{array} { c } 50 \\3\end{array} \right) , \ldots , 50 \left( \begin{array} { c } 50 \\50\end{array} \right) , 0,0,0 , \ldots$

(Short Answer)

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Question 123

Consider the recurrence relation $a _ { n } = 2 a _ { n - 1 } + 1$ (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. (e) Find the particular solution to the given recurrence relation when $a _ { 0 } = 1$

(Short Answer)

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Question 124

Find the solution of the recurrence relation $a _ { n } = 3 a _ { n - 1 } \text { with } a _ { 0 } = 2$

(Short Answer)

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Question 125

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

(Short Answer)

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Question 126

$a _ { n } = a _ { n - 1 } + 2 n + 1 , a _ { 0 } = 5$

(Short Answer)

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Question 127

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 $F ( n ) = n 2 ^ { n }$ ?

(Short Answer)

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Question 128

Find $\mid A _ { 1 } \cup A _ { 2 } \cup A _ { 3 } \cup A _ { 4 }|$ if each set A_i has 100 elements, each intersection of two sets has 60 elements, each intersection of three sets has 20 elements, and there are 10 elements in all four sets.

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

Find the solution of the recurrence relation $a _ { n } = 3 a _ { n - 1 } \text { with } a _ { 0 } = 2$

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

$a _ { n } = a _ { n - 1 } + 2 n + 1 , a _ { 0 } = 5$

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 $F ( n ) = n 2 ^ { n }$ ?

The Foundations: Logic and Proofs

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

Algorithms

Number Theory and Cryptography

Induction and Recursion

Counting

Discrete Probability

Relations

Graphs

Trees

Boolean Algebra

Modeling Computation

Filters

Exam 8: Advanced Counting Techniques

Find ∣A1∪A2∪A3∪A4∣\mid A _ { 1 } \cup A _ { 2 } \cup A _ { 3 } \cup A _ { 4 }|∣A1​∪A2​∪A3​∪A4​∣ if each set Ai has 100 elements, each intersection of two sets has 60 elements, each intersection of three sets has 20 elements, and there are 10 elements in all four sets.

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

Find the solution of the recurrence relation an=3an−1 with a0=2a _ { n } = 3 a _ { n - 1 } \text { with } a _ { 0 } = 2an​=3an−1​ with a0​=2

In questions 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=1,a1=0a _ { n } = 5 a _ { n - 1 } - 4 a _ { n - 2 } , \quad a _ { 0 } = 1 , a _ { 1 } = 0an​=5an−1​−4an−2​,a0​=1,a1​=0

an=an−1+2n+1,a0=5a _ { n } = a _ { n - 1 } + 2 n + 1 , a _ { 0 } = 5an​=an−1​+2n+1,a0​=5

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)=n2nF ( n ) = n 2 ^ { n }F(n)=n2n ?

The Foundations: Logic and Proofs

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

Algorithms

Number Theory and Cryptography

Induction and Recursion

Counting

Discrete Probability

Relations

Graphs

Trees

Boolean Algebra

Modeling Computation

Filters

Find $\mid A _ { 1 } \cup A _ { 2 } \cup A _ { 3 } \cup A _ { 4 }|$ if each set A_i has 100 elements, each intersection of two sets has 60 elements, each intersection of three sets has 20 elements, and there are 10 elements in all four sets.

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

Find the solution of the recurrence relation $a _ { n } = 3 a _ { n - 1 } \text { with } a _ { 0 } = 2$

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

$a _ { n } = a _ { n - 1 } + 2 n + 1 , a _ { 0 } = 5$

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 $F ( n ) = n 2 ^ { n }$ ?