Exam 5: A: Induction and Recursion

Prove that $\frac { 1 } { 2 } + \frac { 2 } { 4 } + \frac { 3 } { 8 } + \cdots + \frac { n } { 2 ^ { n } } = \frac { 2 ^ { n + 1 } - 2 - n } { 2 ^ { n } }$ for all $n \geq 1$ .

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Question 1

Correct Answer:

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P(1): $\frac { 1 } { 2 } = \frac { \left( 2 ^ { 2 } - 2 - 1 \right) } { 2 ^ { 1 } }$ which is true since the right side is equal to 1/2 . $P ( k ) \rightarrow P ( k + 1 ) : \frac { 1 } { 2 } + \frac { 2 } { 4 } + \frac { 3 } { 8 } + \cdots$ +
$\frac { k + 1 } { 2 k + 1 } = \frac { 2 ^ { k + 1 } - 2 - k } { 2 ^ { k } } + \frac { k + 1 } { 2 ^ { k + 1 } } = \frac { 2 ^ { k + 2 } - 4 - 2 k + k + 1 } { 2 k + 1 } = \frac { 2 ^ { k + 2 } - 3 - k } { 2 ^ { k + 1 } } = \frac { 2 ^ { k + 2 } - 2 - ( k + 1 ) } { 2 ^ { k + 1 } }$

Suppose you wish to use mathematical induction to prove that: $1 \cdot 1 ! + 2 \cdot 2 ! + 3 \cdot 3 ! + \cdots + n \cdot n ! = ( n + 1 ) ! - 1 \quad \text { for all } n \geq 1$ (a) Write P(1). (b) Write P(5). (c) Write P(k). (d) Write P(k + 1). (e) Use mathematical induction to prove that P(n) is true for all n ≥ 1.

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Question 2

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(a) $1 \cdot 1 ! = 2 ! - 1$
(b) $1 \cdot 1 ! + 2 \cdot 2 ! + \cdots + 5 \cdot 5 ! = 6 ! - 1$
(c) $1 \cdot 1 ! + 2 \cdot 2 ! + \cdots + k \cdot k ! = ( k + 1 ) ! - 1$
(d) $1 \cdot 1 ! + 2 \cdot 2 ! + \cdots + ( k + 1 ) ( k + 1 ) ! = ( k + 2 ) ! - 1$
(e) $P ( 1 )$ is true since $1 \cdot 1 ! = 1$ and $2 ! - 1 = 1 . \quad P ( k ) \rightarrow P ( k + 1 ) : 1 \cdot 1 ! + 2 \cdot 2 ! + \cdots + ( k + 1 ) ( k + 1 ) ! =$ $( k + 1 ) ! - 1 + ( k + 1 ) ( k + 1 ) ! = ( k + 1 ) ! [ 1 + ( k + 1 ) ] - 1 = ( k + 1 ) ! ( k + 2 ) - 1 = ( k + 2 ) ! - 1 .$

Use mathematical induction to prove that any integer amount of postage from 18 cents on up can be made from an inﬁnite supply of 4-cent and 7-cent stamps.

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Question 3

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$P ( 18 ) :$ use one 4 -cent stamp and two 7-cent stamps. $\quad P ( k ) \rightarrow P ( k + 1 )$ : if a pile of stamps for $k$ cents postage has a 7-cent stamp, replace one 7 -cent stamp with two 4-cent stamps; if the pile contains only 4-cent stamps (there must be at least five of them), replace five 4-cent stamps with three 7-cent stamps.

Suppose {a_n} is deﬁned recursively by an = $a^{2}_{n−1} − 1$ and that a₀ = 2. Find a3 and a4.

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Question 4

Use mathematical induction to prove that $1 - 2 + 2 ^ { 2 } - 2 ^ { 3 } + \cdots + ( - 1 ) ^ { n } 2 ^ { n } = \frac { 2 ^ { n + 1 } ( - 1 ) ^ { n } + 1 } { 3 } \text { for all positive }$ integers n.

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Question 5

give a recursive deﬁnition with initial condition(s). -The function $f ( n ) = n ! , n = 0,1,2 , \ldots$

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Question 6

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 $P ( k ) \rightarrow P ( k + 1 )$ 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 ."

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Question 7

Use mathematical induction to prove that $4 \mid \left( 9 ^ { n } - 5 ^ { n } \right)$ for all $n \geq 0$

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Question 8

give a recursive deﬁnition with initial condition(s). -The function $f ( n ) = 5 n + 2 , n = 1,2,3 , \ldots$

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Question 9

give a recursive deﬁnition with initial condition(s). -The function $f ( n ) = 2 ^ { n } , n = 1,2,3 , \ldots$

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Question 10

Use mathematical induction to prove that $1 + 4 + 7 + 10 + \cdots + ( 3 n - 2 ) = \frac { n ( 3 n - 1 ) } { 2 } \text { for all } n \geq 1$

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Question 11

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

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Question 12

Use mathematical induction to prove that $5 \mid \left( 7 ^ { n } - 2 ^ { n } \right)$ for all $n \geq 0$

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Question 13

give a recursive deﬁnition (with initial condition(s)) of $\left\{ a _ { n } \right\} \quad ( n = 1,2,3 , \ldots )$ - $a _ { n } = 3 n - 5$

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Question 14

Use mathematical induction to prove that $2 \mid \left( n ^ { 2 } + 3 n \right) \text { for all } n \geq 1 \text {. }$

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Question 15

Find f(2) and f(3) if f(n) = f(n − 1) · f(n − 2) + 1, f(0) = 1, f(1) = 4.

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Question 16

give a recursive deﬁnition with initial condition(s) of the set S . -{. . . , −5, −3, −1, 1, 3, 5, . . .}

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Question 17

Prove that the distributive law $A _ { 1 } \cap \left( A _ { 2 } \cup \cdots \cup A _ { n } \right) = \left( A _ { 1 } \cap A _ { 2 } \right) \cup \cdots \cup \left( A _ { 1 } \cap A _ { n } \right)$ is true for all $n \geq 3$

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Question 18

Verify that the program segment a:=2 b:=a+c is correct with respect to the initial assertion c = 3 and the ﬁnal assertion b = 5.

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Question 19

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

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Question 20

Exam 5: A: Induction and Recursion

Prove that 12+24+38+⋯+n2n=2n+1−2−n2n\frac { 1 } { 2 } + \frac { 2 } { 4 } + \frac { 3 } { 8 } + \cdots + \frac { n } { 2 ^ { n } } = \frac { 2 ^ { n + 1 } - 2 - n } { 2 ^ { n } }21​+42​+83​+⋯+2nn​=2n2n+1−2−n​ for all n≥1n \geq 1n≥1 .

Use mathematical induction to prove that any integer amount of postage from 18 cents on up can be made from an inﬁnite supply of 4-cent and 7-cent stamps.

Suppose {an} is deﬁned recursively by an = an−12−1a^{2}_{n−1} − 1an−12​−1 and that a0 = 2. Find a3 and a4.

give a recursive deﬁnition with initial condition(s). -The function f(n)=n!,n=0,1,2,…f ( n ) = n ! , n = 0,1,2 , \ldotsf(n)=n!,n=0,1,2,…

Use mathematical induction to prove that 4∣(9n−5n)4 \mid \left( 9 ^ { n } - 5 ^ { n } \right)4∣(9n−5n) for all n≥0n \geq 0n≥0

give a recursive deﬁnition with initial condition(s). -The function f(n)=5n+2,n=1,2,3,…f ( n ) = 5 n + 2 , n = 1,2,3 , \ldotsf(n)=5n+2,n=1,2,3,…

give a recursive deﬁnition with initial condition(s). -The function f(n)=2n,n=1,2,3,…f ( n ) = 2 ^ { n } , n = 1,2,3 , \ldotsf(n)=2n,n=1,2,3,…

Use mathematical induction to prove that 1+4+7+10+⋯+(3n−2)=n(3n−1)2 for all n≥11 + 4 + 7 + 10 + \cdots + ( 3 n - 2 ) = \frac { n ( 3 n - 1 ) } { 2 } \text { for all } n \geq 11+4+7+10+⋯+(3n−2)=2n(3n−1)​ for all n≥1

give a recursive deﬁnition (with initial condition(s)) of {an}(n=1,2,3,…)\left\{ a _ { n } \right\} \quad ( n = 1,2,3 , \ldots ){an​}(n=1,2,3,…) - an=21/2na _ { n } = 2 ^ { 1 / 2 ^ { n } }an​=21/2n

Use mathematical induction to prove that 5∣(7n−2n)5 \mid \left( 7 ^ { n } - 2 ^ { n } \right)5∣(7n−2n) for all n≥0n \geq 0n≥0

give a recursive deﬁnition (with initial condition(s)) of {an}(n=1,2,3,…)\left\{ a _ { n } \right\} \quad ( n = 1,2,3 , \ldots ){an​}(n=1,2,3,…) - an=3n−5a _ { n } = 3 n - 5an​=3n−5

Use mathematical induction to prove that 2∣(n2+3n) for all n≥1. 2 \mid \left( n ^ { 2 } + 3 n \right) \text { for all } n \geq 1 \text {. }2∣(n2+3n) for all n≥1.

Find f(2) and f(3) if f(n) = f(n − 1) · f(n − 2) + 1, f(0) = 1, f(1) = 4.

give a recursive deﬁnition with initial condition(s) of the set S . -{. . . , −5, −3, −1, 1, 3, 5, . . .}

Verify that the program segment a:=2 b:=a+c is correct with respect to the initial assertion c = 3 and the ﬁnal assertion b = 5.

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

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

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

Prove that $\frac { 1 } { 2 } + \frac { 2 } { 4 } + \frac { 3 } { 8 } + \cdots + \frac { n } { 2 ^ { n } } = \frac { 2 ^ { n + 1 } - 2 - n } { 2 ^ { n } }$ for all $n \geq 1$ .

Suppose {a_n} is deﬁned recursively by an = $a^{2}_{n−1} − 1$ and that a₀ = 2. Find a3 and a4.

give a recursive deﬁnition with initial condition(s). -The function $f ( n ) = n ! , n = 0,1,2 , \ldots$

Use mathematical induction to prove that $4 \mid \left( 9 ^ { n } - 5 ^ { n } \right)$ for all $n \geq 0$

give a recursive deﬁnition with initial condition(s). -The function $f ( n ) = 5 n + 2 , n = 1,2,3 , \ldots$

give a recursive deﬁnition with initial condition(s). -The function $f ( n ) = 2 ^ { n } , n = 1,2,3 , \ldots$

Use mathematical induction to prove that $1 + 4 + 7 + 10 + \cdots + ( 3 n - 2 ) = \frac { n ( 3 n - 1 ) } { 2 } \text { for all } n \geq 1$

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

Use mathematical induction to prove that $5 \mid \left( 7 ^ { n } - 2 ^ { n } \right)$ for all $n \geq 0$

give a recursive deﬁnition (with initial condition(s)) of $\left\{ a _ { n } \right\} \quad ( n = 1,2,3 , \ldots )$ - $a _ { n } = 3 n - 5$

Use mathematical induction to prove that $2 \mid \left( n ^ { 2 } + 3 n \right) \text { for all } n \geq 1 \text {. }$