Question 1

Find  f(2)  and  f(3)  if $f ( n ) = f ( n - 1 ) \cdot f ( n - 2 ) + 1$  f(0)=1, f(1)=4

Accepted Answer

f(2)=5, f(3)=21

Question 2

Use the Principle of Mathematical Induction to prove that $$4 \mid \left( 9 ^ { n } - 5 ^ { n } \right) \text { for all } n \geq 0$$

Accepted Answer

$P ( 0 ) : 4 \mid 1 - 1$ is true since $4 \mid 0 . \quad P ( k ) \rightarrow P ( k + 1 ) : 9 ^ { k + 1 } - 5 ^ { k + 1 } = 9 \left( 9 ^ { k } - 5 ^ { k } \right) + 5 ^ { k } ( 9 - 5 )$. Each term is divisible by $4 : 4 \rfloor 9 ^ { k } - 5 ^ { k } ($ bv $P ( k ) )$ and 4 $\rfloor 9 - 5$,

Question 3

Give a recursive algorithm for computing $$n a \text {, where } n \text { is a positive integer and } a \text { is a real number. }$$

Accepted Answer

The following procedure computes na: procedure mult( $a$ : real number, $n$ : positive integer)
if $n = 1$ then mult $( a , n ) : = a$
else $\operatorname { mult } ( a , n ) : = a + \operatorname { mult } ( a , n - 1 )$.

Question 4

Use the Principle of Mathematical Induction to prove that $$1 + 3 + 9 + 27 + \cdots + 3 ^ { n } = \frac { 3 ^ { n + 1 } - 1 } { 2 } \text { for all } n \geq 0$$

Accepted Answer

The answer of Use the Principle of Mathematical Induction to...

Question 5

Describe a recursive algorithm for computing $$3 ^ { 3 ^ { n } } \text { where } n \text { is a nonnegative integer. }$$

Accepted Answer

The answer of Describe a recursive algorithm for computing \[3...

Question 6

Use the Principle of Mathematical Induction to prove that $$1 + 4 + 7 + 10 + \cdots + ( 3 n - 2 ) = \frac { n ( 3 n - 1 ) } { 2 } \text { for all }$$ n ≥ 1.

Accepted Answer

The answer of Use the Principle of Mathematical Induction to...

Question 7

Suppose $$\left\{ a _ { n } \right\} \text { is defined recursively by } a _ { n } = a _ { n - 1 } ^ { 2 } - 1 \text { and that } a _ { 0 } = 2 \text {. Find } a _ { 3 } \text { and } a _ { 4 } \text {. }$$

Accepted Answer

The answer of Suppose \[\left\{ a _ { n }...

Question 8

Find the error in the following proof of this "theorem":
"Theorem: Every positive integer equals the next largest positive integer."
"Proof: Let  P(n)  be the proposition $' n = n + 1 '$
To show that $P ( k ) \rightarrow P ( k + 1 )$ assume that  P(k)  is true for some  k , so that  k=k+1 . Add 1 to both sides of this equation to obtain  k+1=k+2 , which is  P(k+1) . Therefore $P ( k ) \rightarrow P ( k + 1 )$ is true. Hence  P(n)  is true for all positive integers  n . "

Accepted Answer

No basis c

Question 9

Find  f(2)  and  f(3)  if $f ( n ) = 2 f ( n - 1 ) + 6 , f ( 0 ) = 3$

Accepted Answer

The answer of Find  f(2)  and  f(3)...

Question 10

Use the Principle of Mathematical Induction to prove that $$5 \mid \left( 7 ^ { n } - 2 ^ { n } \right) \text { for all } n \geq 0$$

Accepted Answer

The answer of Use the Principle of Mathematical Induction to...

Question 11

Floor borders one foot wide and of varying lengths are to be covered with nonoverlapping tiles that are available in two sizes: $1 ^ { \prime } \times 3 ^ { \prime } \text { and } 1 ^ { \prime } \times 5 ^ { \prime }$ sizes. Assuming that the supply of each size is infinite, prove that every $1 ^ { \prime } \times n ^ { \prime }$ border  (n > 7)  can be covered with these tiles.

Accepted Answer

P(8):  use one of each type. 11ec7ce0_d9

Question 12

Use the Principle of Mathematical Induction to prove that $$2 n + 3 \leq 2 ^ { n } \text { for all } n \geq 4 \text {. }$$

Accepted Answer

The answer of Use the Principle of Mathematical Induction to...

Question 13

In questions give a recursive deﬁnition with initial condition(s) of the set S.
-$$\text { The set of strings } 1,111,11111,1111111 , \ldots$$

Accepted Answer

The answer of In questions give a recursive deﬁnition with...

Question 14

give a recursive deﬁnition (with initial condition(s)) of $$\left\{ a _ { n } \right\} ( n = 1,2,3 , \ldots )$$
-$$a _ { n } = 2 ^ { n }$$

Accepted Answer

The answer of give a recursive deﬁnition (with initial condition(s))...

Question 15

Use the Principle of Mathematical Induction to prove that $$1 + 2 ^ { n } \leq 3 ^ { n } \text { for all } n \geq 1$$

Accepted Answer

The answer of Use the Principle of Mathematical Induction to...

Question 16

Use the Principle of Mathematical Induction to prove that $$n ^ { 3 } > n ^ { 2 } + 3 \text { for all } n \geq 2$$

Accepted Answer

The answer of Use the Principle of Mathematical Induction to...

Question 17

Use the Principle of Mathematical Induction to prove that $1 - 2 + 2 ^ { 2 } - 2 ^ { 3 } + \cdots + ( - 1 ) ^ { n } 2 ^ { n } = \frac { 2 ^ { n + 1 } ( - 1 ) ^ { n } + 1 } { 3 }$ for all positive integers  n .

Accepted Answer

The answer of Use the Principle of Mathematical Induction to...

Question 18

Use the Principle of Mathematical Induction to prove that $$2 \mid \left( n ^ { 2 } + 3 n \right) \text { for all } n \geq 1$$

Accepted Answer

The answer of Use the Principle of Mathematical Induction to...

Question 19

Suppose that the only paper money consists of 3-dollar bills and 10-dollar bills. Show that any dollar amount
greater than 17 dollars could be made from a combination of these bills.

Accepted Answer

The answer of Suppose that the only paper money consists...

Question 20

In questions give a recursive deﬁnition with initial condition(s) of the set S.
-$$\{ 0.1,0.01,0.001,0.0001 \}$$

Accepted Answer

The answer of In questions give a recursive deﬁnition with...

Find f(2) and f(3) if $f ( n ) = f ( n - 1 ) \cdot f ( n - 2 ) + 1$ f(0)=1, f(1)=4

Use the Principle of Mathematical Induction to prove that $4 \mid \left( 9 ^ { n } - 5 ^ { n } \right) \text { for all } n \geq 0$

Give a recursive algorithm for computing $n a \text {, where } n \text { is a positive integer and } a \text { is a real number. }$

Use the Principle of Mathematical Induction to prove that $1 + 3 + 9 + 27 + \cdots + 3 ^ { n } = \frac { 3 ^ { n + 1 } - 1 } { 2 } \text { for all } n \geq 0$

Describe a recursive algorithm for computing $3 ^ { 3 ^ { n } } \text { where } n \text { is a nonnegative integer. }$

Use the Principle of Mathematical Induction to prove that $1 + 4 + 7 + 10 + \cdots + ( 3 n - 2 ) = \frac { n ( 3 n - 1 ) } { 2 } \text { for all }$ n ≥ 1.

Suppose $\left\{ a _ { n } \right\} \text { is defined recursively by } a _ { n } = a _ { n - 1 } ^ { 2 } - 1 \text { and that } a _ { 0 } = 2 \text {. Find } a _ { 3 } \text { and } a _ { 4 } \text {. }$

Find f(2) and f(3) if $f ( n ) = 2 f ( n - 1 ) + 6 , f ( 0 ) = 3$

Use the Principle of Mathematical Induction to prove that $5 \mid \left( 7 ^ { n } - 2 ^ { n } \right) \text { for all } n \geq 0$

Use the Principle of Mathematical Induction to prove that $2 n + 3 \leq 2 ^ { n } \text { for all } n \geq 4 \text {. }$

In questions give a recursive deﬁnition with initial condition(s) of the set S. - $\text { The set of strings } 1,111,11111,1111111 , \ldots$

give a recursive deﬁnition (with initial condition(s)) of $\left\{ a _ { n } \right\} ( n = 1,2,3 , \ldots )$ - $a _ { n } = 2 ^ { n }$

Use the Principle of Mathematical Induction to prove that $1 + 2 ^ { n } \leq 3 ^ { n } \text { for all } n \geq 1$

Use the Principle of Mathematical Induction to prove that $n ^ { 3 } > n ^ { 2 } + 3 \text { for all } n \geq 2$

Use the Principle of Mathematical Induction to prove that $1 - 2 + 2 ^ { 2 } - 2 ^ { 3 } + \cdots + ( - 1 ) ^ { n } 2 ^ { n } = \frac { 2 ^ { n + 1 } ( - 1 ) ^ { n } + 1 } { 3 }$ for all positive integers n .

Use the Principle of Mathematical Induction to prove that $2 \mid \left( n ^ { 2 } + 3 n \right) \text { for all } n \geq 1$

Suppose that the only paper money consists of 3-dollar bills and 10-dollar bills. Show that any dollar amount greater than 17 dollars could be made from a combination of these bills.

In questions give a recursive deﬁnition with initial condition(s) of the set S. - $\{ 0.1,0.01,0.001,0.0001 \}$

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Exam 5: Induction and Recursion

Find f(2) and f(3) if f(n)=f(n−1)⋅f(n−2)+1f ( n ) = f ( n - 1 ) \cdot f ( n - 2 ) + 1f(n)=f(n−1)⋅f(n−2)+1 f(0)=1, f(1)=4

Use the Principle of Mathematical Induction to prove that 4∣(9n−5n) for all n≥04 \mid \left( 9 ^ { n } - 5 ^ { n } \right) \text { for all } n \geq 04∣(9n−5n) for all n≥0

Give a recursive algorithm for computing na, where n is a positive integer and a is a real number. n a \text {, where } n \text { is a positive integer and } a \text { is a real number. }na, where n is a positive integer and a is a real number.

Use the Principle of Mathematical Induction to prove that 1+3+9+27+⋯+3n=3n+1−12 for all n≥01 + 3 + 9 + 27 + \cdots + 3 ^ { n } = \frac { 3 ^ { n + 1 } - 1 } { 2 } \text { for all } n \geq 01+3+9+27+⋯+3n=23n+1−1​ for all n≥0

Describe a recursive algorithm for computing 33n where n is a nonnegative integer. 3 ^ { 3 ^ { n } } \text { where } n \text { is a nonnegative integer. }33n where n is a nonnegative integer.

Use the Principle of Mathematical Induction to prove that 1+4+7+10+⋯+(3n−2)=n(3n−1)2 for all 1 + 4 + 7 + 10 + \cdots + ( 3 n - 2 ) = \frac { n ( 3 n - 1 ) } { 2 } \text { for all }1+4+7+10+⋯+(3n−2)=2n(3n−1)​ for all n ≥ 1.

Find f(2) and f(3) if f(n)=2f(n−1)+6,f(0)=3f ( n ) = 2 f ( n - 1 ) + 6 , f ( 0 ) = 3f(n)=2f(n−1)+6,f(0)=3

Use the Principle of Mathematical Induction to prove that 5∣(7n−2n) for all n≥05 \mid \left( 7 ^ { n } - 2 ^ { n } \right) \text { for all } n \geq 05∣(7n−2n) for all n≥0

Use the Principle of Mathematical Induction to prove that 2n+3≤2n for all n≥4. 2 n + 3 \leq 2 ^ { n } \text { for all } n \geq 4 \text {. }2n+3≤2n for all n≥4.

In questions give a recursive deﬁnition with initial condition(s) of the set S. - The set of strings 1,111,11111,1111111,…\text { The set of strings } 1,111,11111,1111111 , \ldots The set of strings 1,111,11111,1111111,…

give a recursive deﬁnition (with initial condition(s)) of {an}(n=1,2,3,…)\left\{ a _ { n } \right\} ( n = 1,2,3 , \ldots ){an​}(n=1,2,3,…) - an=2na _ { n } = 2 ^ { n }an​=2n

Use the Principle of Mathematical Induction to prove that 1+2n≤3n for all n≥11 + 2 ^ { n } \leq 3 ^ { n } \text { for all } n \geq 11+2n≤3n for all n≥1

Use the Principle of Mathematical Induction to prove that n3>n2+3 for all n≥2n ^ { 3 } > n ^ { 2 } + 3 \text { for all } n \geq 2n3>n2+3 for all n≥2

Use the Principle of Mathematical Induction to prove that 1−2+22−23+⋯+(−1)n2n=2n+1(−1)n+131 - 2 + 2 ^ { 2 } - 2 ^ { 3 } + \cdots + ( - 1 ) ^ { n } 2 ^ { n } = \frac { 2 ^ { n + 1 } ( - 1 ) ^ { n } + 1 } { 3 }1−2+22−23+⋯+(−1)n2n=32n+1(−1)n+1​ for all positive integers n .

Use the Principle of Mathematical Induction to prove that 2∣(n2+3n) for all n≥12 \mid \left( n ^ { 2 } + 3 n \right) \text { for all } n \geq 12∣(n2+3n) for all n≥1

Suppose that the only paper money consists of 3-dollar bills and 10-dollar bills. Show that any dollar amount greater than 17 dollars could be made from a combination of these bills.

In questions give a recursive deﬁnition with initial condition(s) of the set S. - {0.1,0.01,0.001,0.0001}\{ 0.1,0.01,0.001,0.0001 \}{0.1,0.01,0.001,0.0001}

The Foundations: Logic and Proofs

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

Algorithms

Number Theory and Cryptography

Counting

Discrete Probability

Advanced Counting Techniques

Relations

Graphs

Trees

Boolean Algebra

Modeling Computation

Filters

Find f(2) and f(3) if $f ( n ) = f ( n - 1 ) \cdot f ( n - 2 ) + 1$ f(0)=1, f(1)=4

Use the Principle of Mathematical Induction to prove that $4 \mid \left( 9 ^ { n } - 5 ^ { n } \right) \text { for all } n \geq 0$

Give a recursive algorithm for computing $n a \text {, where } n \text { is a positive integer and } a \text { is a real number. }$

Use the Principle of Mathematical Induction to prove that $1 + 3 + 9 + 27 + \cdots + 3 ^ { n } = \frac { 3 ^ { n + 1 } - 1 } { 2 } \text { for all } n \geq 0$

Describe a recursive algorithm for computing $3 ^ { 3 ^ { n } } \text { where } n \text { is a nonnegative integer. }$

Use the Principle of Mathematical Induction to prove that $1 + 4 + 7 + 10 + \cdots + ( 3 n - 2 ) = \frac { n ( 3 n - 1 ) } { 2 } \text { for all }$ n ≥ 1.

Find f(2) and f(3) if $f ( n ) = 2 f ( n - 1 ) + 6 , f ( 0 ) = 3$

Use the Principle of Mathematical Induction to prove that $5 \mid \left( 7 ^ { n } - 2 ^ { n } \right) \text { for all } n \geq 0$

Use the Principle of Mathematical Induction to prove that $2 n + 3 \leq 2 ^ { n } \text { for all } n \geq 4 \text {. }$

In questions give a recursive deﬁnition with initial condition(s) of the set S. - $\text { The set of strings } 1,111,11111,1111111 , \ldots$

give a recursive deﬁnition (with initial condition(s)) of $\left\{ a _ { n } \right\} ( n = 1,2,3 , \ldots )$ - $a _ { n } = 2 ^ { n }$

Use the Principle of Mathematical Induction to prove that $1 + 2 ^ { n } \leq 3 ^ { n } \text { for all } n \geq 1$

Use the Principle of Mathematical Induction to prove that $n ^ { 3 } > n ^ { 2 } + 3 \text { for all } n \geq 2$

Use the Principle of Mathematical Induction to prove that $1 - 2 + 2 ^ { 2 } - 2 ^ { 3 } + \cdots + ( - 1 ) ^ { n } 2 ^ { n } = \frac { 2 ^ { n + 1 } ( - 1 ) ^ { n } + 1 } { 3 }$ for all positive integers n .

Use the Principle of Mathematical Induction to prove that $2 \mid \left( n ^ { 2 } + 3 n \right) \text { for all } n \geq 1$

In questions give a recursive deﬁnition with initial condition(s) of the set S. - $\{ 0.1,0.01,0.001,0.0001 \}$