Deck 5: Induction and Recursion

Question 1

Prove that

Answer 1

Question 2

Use the Principle of Mathematical Induction to prove that

Answer 2

Question 3

Use the Principle of Mathematical Induction to prove that for all positive integers n .

Answer 3

Question 4

Use the Principle of Mathematical Induction to prove that

Question 5

Suppose you wish to prove that the following is true for all positive integers n by using the Principle of Mathematical Induction: 1+3+5+...+(2 n-1)=n² .
(a) Write P(1)
(b) Write P(72)
(c) Write P(73)
(d) Use P(72) to prove P(73)
(e) Write P(k)
(f) Write P(k+1)
(g) Use the Principle of Mathematical Induction to prove that P(n) is true for all positive integers n

Question 6

Use the Principle of Mathematical Induction to prove that n ≥ 1.

Question 7

Suppose you wish to use the Principle of Mathematical Induction to prove that
(a) Write P(1)
(b) Write P(5)
(c) Write P(k)
(d) Write P(k+1)
(e) Use the Principle of Mathematical Induction to prove that P(n) is true for all

Question 8

Use the Principle of Mathematical Induction to prove that

Question 9

A T -omino is a tile pictured at the right. Prove that every chessboard can be tiled with T-ominoes.

Question 10

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.

Question 11

Use the Principle of Mathematical Induction to prove that

Question 12

Use mathematical induction to prove that every integer amount of postage of six cents or more can be formed
using 3-cent and 4-cent stamps.

Question 13

Use the Principle of Mathematical Induction to prove that

Question 14

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

Question 15

Use the Principle of Mathematical Induction to prove that

Question 16

Floor borders one foot wide and of varying lengths are to be covered with nonoverlapping tiles that are available in two sizes: sizes. Assuming that the supply of each size is infinite, prove that every border (n > 7) can be covered with these tiles.

Question 17

Use the Principle of Mathematical Induction to prove that

Question 18

Use the Principle of Mathematical Induction to prove that

Question 19

Let

Question 20

Use the Principle of 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.

Question 21

In questions give a recursive definition with initial condition(s).
The sequence

Question 22

In questions give a recursive definition with initial condition(s).
The set

Question 23

In questions give a recursive definition with initial condition(s).
The function

Question 24

give a recursive definition (with initial condition(s)) of

Question 25

Prove that for all

Question 26

In questions give a recursive definition with initial condition(s).
The function

Question 27

give a recursive definition (with initial condition(s)) of

Question 28

In questions give a recursive definition with initial condition(s) of the set S.

Question 29

give a recursive definition (with initial condition(s)) of

Question 30

give a recursive definition (with initial condition(s)) of

Question 31

In questions give a recursive definition with initial condition(s).
The Fibonacci numbers 1, 1, 2, 3, 5, 8, 13, . . . .

Question 32

give a recursive definition (with initial condition(s)) of

Question 33

Use the Principle of Mathematical Induction to prove that

Question 34

In questions give a recursive definition with initial condition(s).
The set

Question 35

Prove that all distributive law is true for all

Question 36

give a recursive definition (with initial condition(s)) of

Question 37

In questions give a recursive definition with initial condition(s).
The function

Question 38

In questions give a recursive definition with initial condition(s).
The set

Question 39

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
To show that 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 is true. Hence P(n) is true for all positive integers n . "

Question 40

In questions give a recursive definition with initial condition(s).
The set

Question 41

In questions give a recursive definition with initial condition(s) of the set S.

Question 42

In questions give a recursive definition with initial condition(s) of the set S.

Question 43

Find f(2) and f(3) if

Question 44

Describe a recursive algorithm for computing

Question 45

Consider the following program segment: invariant.

Question 46

Give a recursive algorithm for computing

Question 47

Verify that the following program segment is correct with respect to the initial assertion T and the final
assertion (x ≤ y ∧ max = y) ∨ (x > y ∧ max = x):
if x ≤ y then
max := y
else
max := x

Question 48

In questions give a recursive definition with initial condition(s) of the set S.
All positive integer multiples of 5.

Question 49

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

Question 50

Find f(2) and f(3) if f(n)=f(n-1) / f(n-2), f(0)=2, f(1)=5 .

Question 51

Find f(2) and f(3) if f(0)=1, f(1)=4

Question 52

Suppose

Question 53

In questions give a recursive definition with initial condition(s) of the set S.