IB Mathematics Analysis and Approaches Topic AHL 1.15—proof by Induction, Contradiction, Counterexamples Revision Notes (HL)

IB Mathematics Analysis and Approaches Topic AHL 1.15—proof by Induction, Contradiction, Counterexamples Revision Notes (HL) | RevisionDojo

Written by official IB examiners

Proof by Mathematical Induction

Mathematical induction is a method of proof used to establish that a statement is true for all natural numbers. The main idea is that you assume that the statement is true for some natural number. Then you show that if it is true for that natural number, then it is true for the next. Hence, if you show it is true for one number, then it is true for the next, and so on, proving it is true for all numbers greater than or equal to the base case.

Steps

It consists of a few main steps (and one extra but important step for the examinations):

Call your statement $P(n)$. For example, "$P(n)$: $n(n+1)(n+2)$ is divisible by $6$ for all numbers $n\in\mathbb{N}$."
Base case: Prove the statement is true for the smallest value (usually $n = 1$ or $n = 0$, but there can be others, so make sure to check what the smallest possible value is). Call this $P(1)$ or $P(0)$ (or whatever number you're using as the base case).
Inductive hypothesis: Assume $P(k)$ holds for some number $k\in\mathbb{N}$, i.e. the statement is true for some arbitrary natural number $k$.e. the statement is true for some arbitrary natural number $k$. Let this $k$ be bigger than or equal to the number you used in your base step.

Nice try, unfortunately this paywall isn't as easy to bypass as you think. Want to help devleop the site? Join the team at https://revisiondojo.com/join-us. exercitation voluptate cillum ullamco excepteur sint officia do tempor Lorem irure minim Lorem elit id voluptate reprehenderit voluptate laboris in nostrud qui non Lorem nostrud laborum culpa sit occaecat reprehenderit

Definition

Paywall

(on a website) an arrangement whereby access is restricted to users who have paid to subscribe to the site.

anim nostrud sit dolore minim proident quis fugiat velit et eiusmod nulla quis nulla mollit dolor sunt culpa aliqua

Lorem ipsum dolor sit amet, consectetur adipiscing elit. Sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris nisi ut aliquip ex ea commodo consequat.

Duis aute irure dolor in reprehenderit

Duis aute irure dolor in reprehenderit in voluptate velit esse cillum dolore eu fugiat nulla pariatur. Excepteur sint occaecat cupidatat non proident, sunt in culpa qui officia deserunt mollit anim id est laborum.

Note

Lorem ipsum dolor sit amet, consectetur adipiscing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam quis nostrud exercitation.

Excepteur sint occaecat cupidatat non proident

Nemo enim ipsam voluptatem quia voluptas sit aspernatur aut odit aut fugit, sed quia consequuntur magni dolores eos qui ratione voluptatem sequi nesciunt. Neque porro quisquam est, qui dolorem ipsum quia dolor sit amet, consectetur, adipisci velit.

Tip

Lorem ipsum dolor sit amet, consectetur adipiscing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris nisi ut aliquip ex ea commodo consequat.

Lorem ipsum dolor sit amet, consectetur adipiscing elit.
Sed do eiusmod tempor incididunt ut labore et dolore magna aliqua.
Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris.
Duis aute irure dolor in reprehenderit in voluptate velit esse cillum.

Unlock the rest of this chapter with a free account

End of article

Proof by Mathematical Induction

Steps

It consists of a few main steps (and one extra but important step for the examinations):

Call your statement $P(n)$. For example, "$P(n)$: $n(n+1)(n+2)$ is divisible by $6$ for all numbers $n\in\mathbb{N}$."
Base case: Prove the statement is true for the smallest value (usually $n = 1$ or $n = 0$, but there can be others, so make sure to check what the smallest possible value is). Call this $P(1)$ or $P(0)$ (or whatever number you're using as the base case).
Inductive hypothesis: Assume $P(k)$ holds for some number $k\in\mathbb{N}$, i.e. the statement is true for some arbitrary natural number $k$.e. the statement is true for some arbitrary natural number $k$. Let this $k$ be bigger than or equal to the number you used in your base step.

Definition

Paywall

(on a website) an arrangement whereby access is restricted to users who have paid to subscribe to the site.

anim nostrud sit dolore minim proident quis fugiat velit et eiusmod nulla quis nulla mollit dolor sunt culpa aliqua

Duis aute irure dolor in reprehenderit

Note

Lorem ipsum dolor sit amet, consectetur adipiscing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam quis nostrud exercitation.

Excepteur sint occaecat cupidatat non proident

Tip

Lorem ipsum dolor sit amet, consectetur adipiscing elit.
Sed do eiusmod tempor incididunt ut labore et dolore magna aliqua.
Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris.
Duis aute irure dolor in reprehenderit in voluptate velit esse cillum.

Unlock the rest of this chapter with a free account

End of article

Number and Algebra16 subtopics

Functions16 subtopics

Geometry & Trigonometry18 subtopics

Statistics & Probability14 subtopics

Calculus19 subtopics

IB Mathematics Analysis and Approaches AHL 1.15—proof by Induction, Contradiction, Counterexamples Notes

Proof by Mathematical Induction

Steps

anim nostrud sit dolore minim proident quis fugiat velit et eiusmod nulla quis nulla mollit dolor sunt culpa aliqua

Duis aute irure dolor in reprehenderit

Excepteur sint occaecat cupidatat non proident

Unlock the rest of this chapter with a free account

Proof by Mathematical Induction

Steps

anim nostrud sit dolore minim proident quis fugiat velit et eiusmod nulla quis nulla mollit dolor sunt culpa aliqua

Duis aute irure dolor in reprehenderit

Excepteur sint occaecat cupidatat non proident

Unlock the rest of this chapter with a free account

Number and Algebra16 subtopics

Functions16 subtopics

Geometry & Trigonometry18 subtopics

Statistics & Probability14 subtopics

Calculus19 subtopics