Question 1

$$\text { Prove that } \sum _ { j = n } ^ { 2 n - 1 } ( 2 j + 1 ) = 3 n ^ { 2 } \text { whenever } n \text { is a positive integer. }$$

Accepted Answer

The basis step holds since $\sum _ { j = 1 } ^ { 1 } ( 2 j + 1 ) = 3 = 3 \cdot 1 ^ { 2 }$. For the inductive step assume that $\sum _ { j = k } ^ { 2 k - 1 } ( 2 j + 1 ) = 3 k ^ { 2 }$. It follows that $\sum _ { j = k + 1 } ^ { 2 ( k + 1 ) - 1 } ( 2 j + 1 ) = \sum _ { j = k } ^ { 2 k - 1 } ( 2 j + 1 ) - ( 2 k + 1 ) + ( 4 k + 1 ) + ( 4 k + 3 ) = 3 k ^ { 2 } + 6 k + 3 = 3 ( k + 1 ) ^ { 2 }$. This completes the proof.

Question 2

$$\text { Describe a recursive algorithm for computing } 3 ^ { 2 ^ { n } } \text { where } n \text { is a nonnegative integer. }$$

Accepted Answer

We can use the following recursive procedure. procedure $x ( n :$ nonnegative integer $)$
if $n = 0$ then return 3
else return $x ( n - 1 ) \cdot x ( n - 1 )$

Question 3

What is wrong with the following proof that every positive integer equals the next larger positive integer? "Proof." Let $P ( n )$ be the proposition that $n = n + 1$. Assume that $P ( k )$ is true, so that $k = k + 1$. Add 1 to both sides of this equation to obtain $k + 1 = k + 2$. Since this is the statement $P ( k + 1 )$, it follows that $P ( n )$ is true for all positive integers $n$.

Accepted Answer

The error is that no basis step has been done.

Question 4

Suppose that the only currency were 3-dollar bills and 10-dollar bills. Show that every amount greater than 17 dollars could be made from a combination of these bills.

Accepted Answer

We find that 18 dollars can be made usin

Question 5

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

Accepted Answer

The answer of \[\text { Suppose that } \left\{ a...

Question 6

$$\text { Use mathematical induction to show that } \sum _ { j = 0 } ^ { n } ( j + 1 ) = ( n + 1 ) ( n + 2 ) / 2 \text { whenever } n \text { is a nonnegative integer. }$$

Accepted Answer

The basis step follows since @#LAT-DLM&. For the i

Question 7

Give a recursive algorithm for computing na using addition, where n is a positive integer and a is a real

Accepted Answer

We can compute na recursively

Question 8

Use mathematical induction to show that $n$ lines in the plane passing through the same point divide the plane into $2 n$ parts.

Accepted Answer

The basis step follows since one line di

Question 9

$$\text { Show that } 3 ^ { n } < n \text { ! whenever } n \text { is an integer with } n \geq 7 \text {. }$$

Accepted Answer

The basis step holds since @#LAT-DLM&. F

Question 10

$$\text { Let } a _ { 1 } = 2 , a _ { 2 } = 9 \text {, and } a _ { n } = 2 a _ { n - 1 } + 3 a _ { n - 2 } \text { for } n \geq 3 \text {. Show that } a _ { n } \leq 3 ^ { n } \text { for all positive integers } n \text {. }$$

Accepted Answer

Let @#LAT-DLM& be the proposition that @#LAT-DLM&. The proof

$\text { Prove that } \sum _ { j = n } ^ { 2 n - 1 } ( 2 j + 1 ) = 3 n ^ { 2 } \text { whenever } n \text { is a positive integer. }$

$\text { Describe a recursive algorithm for computing } 3 ^ { 2 ^ { n } } \text { where } n \text { is a nonnegative integer. }$

Suppose that the only currency were 3-dollar bills and 10-dollar bills. Show that every amount greater than 17 dollars could be made from a combination of these bills.

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

$\text { Use mathematical induction to show that } \sum _ { j = 0 } ^ { n } ( j + 1 ) = ( n + 1 ) ( n + 2 ) / 2 \text { whenever } n \text { is a nonnegative integer. }$

Give a recursive algorithm for computing na using addition, where n is a positive integer and a is a real

Use mathematical induction to show that $n$ lines in the plane passing through the same point divide the plane into $2 n$ parts.

$\text { Show that } 3 ^ { n } < n \text { ! whenever } n \text { is an integer with } n \geq 7 \text {. }$

$\text { Let } a _ { 1 } = 2 , a _ { 2 } = 9 \text {, and } a _ { n } = 2 a _ { n - 1 } + 3 a _ { n - 2 } \text { for } n \geq 3 \text {. Show that } a _ { n } \leq 3 ^ { n } \text { for all positive integers } n \text {. }$

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

A: Induction and Recursion

Counting

A: Counting

Discrete Probability

A: Discrete Probability

Advanced Counting Techniques

A: 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

Exam 5: Induction and Recursion

Prove that ∑j=n2n−1(2j+1)=3n2 whenever n is a positive integer. \text { Prove that } \sum _ { j = n } ^ { 2 n - 1 } ( 2 j + 1 ) = 3 n ^ { 2 } \text { whenever } n \text { is a positive integer. } Prove that ∑j=n2n−1​(2j+1)=3n2 whenever n is a positive integer.

Describe a recursive algorithm for computing 32n where n is a nonnegative integer. \text { Describe a recursive algorithm for computing } 3 ^ { 2 ^ { n } } \text { where } n \text { is a nonnegative integer. } Describe a recursive algorithm for computing 32n where n is a nonnegative integer.

Suppose that the only currency were 3-dollar bills and 10-dollar bills. Show that every amount greater than 17 dollars could be made from a combination of these bills.

Give a recursive algorithm for computing na using addition, where n is a positive integer and a is a real

Use mathematical induction to show that nnn lines in the plane passing through the same point divide the plane into 2n2 n2n parts.

Show that 3n<n ! whenever n is an integer with n≥7. \text { Show that } 3 ^ { n } < n \text { ! whenever } n \text { is an integer with } n \geq 7 \text {. } Show that 3n<n ! whenever n is an integer with n≥7.

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

A: Induction and Recursion

Counting

A: Counting

Discrete Probability

A: Discrete Probability

Advanced Counting Techniques

A: 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

$\text { Prove that } \sum _ { j = n } ^ { 2 n - 1 } ( 2 j + 1 ) = 3 n ^ { 2 } \text { whenever } n \text { is a positive integer. }$

$\text { Describe a recursive algorithm for computing } 3 ^ { 2 ^ { n } } \text { where } n \text { is a nonnegative integer. }$

Use mathematical induction to show that $n$ lines in the plane passing through the same point divide the plane into $2 n$ parts.

$\text { Show that } 3 ^ { n } < n \text { ! whenever } n \text { is an integer with } n \geq 7 \text {. }$