Proofs pt.2

Question 1

What is the basic principle of mathmatical induction?

Answer 1

If the pattern holds true for one row/term then it must hold for the next row/term. And this means that it will hold for all rows/terms

Question 2

Just turn the page lil bro

Answer 2

Question 3

What are the two steps in in mathmatical induction?

Answer 3

Basis step and inductive step

Question 4

Answer 4

Question 5

What is a closed form?

Answer 5

A form where a sum of variable terms equals an expression that does not contain an elipsis or a summation symbol

Question 6

What formula would you use to calculate the sum from 1 to n (n is an integer)

Answer 6

(n(n+1))/2

It comes from the fact that the above is true

Question 7

Prove this:

Answer 7

Question 8

True or false?

fix this

Answer 8

False. Note a false hypothesis (first line) can create a true conclusion (last line)

Question 9

Answer 9

Question 10

Prove by mathmatical induction that any integer greater than 1 is divisible by a prime number

Answer 10

Question 11

Answer 11

Question 12

What is the well-ordering principle

Answer 12

If a set of integers, S contains one or more integers, all of which are greater than some fixed integer (within the set). Then S, contains a least element.

Question 13

In the following case, if the set has a least element, state what it is. If not, explain why the well-ordering principle is not violated?
“The set of all positive numbers”

Answer 13

There is no least positive real number. For if x is any positive real number, then x/2 is a positive real number that is less than x. No violation of the well-ordering principle occurs because the well-odering principle refers only to sets of integers, and this set is not a set of integers.

Question 14

In the following case, if the set has a least element, state what it is. If not, explain why the well-ordering principle is not violated?
“The set of all nonnegative integers n such that n² < n”

Answer 14

There is no least nonnegative inter such that n² because there is no nonnegative integer that satisfies this inequality. The well-ordering principle is not violated because it only refers to sets that contain at least one element.

Question 15

In the following case, if the set has a least element, state what it is. If not, explain why the well-ordering principle is not violated?
“The set of all nonnegative integers of the form 46-7k where k is an intger”

Answer 15

Question 16

Answer 16

Question 17

What is it called when you find an explicit formula of a recurrence relation?

Answer 17

You find the solution to the reccurence relation

Question 18

How do you solve recurrence relations by iteration?

Answer 18

You find the first few terms and guess an explict formula

Question 19

Proof the following:

Answer 19