Mathematical induction

Main post link -> https://brilliant.org/assessment/techniques-trainer/mathematical-induction/

Learn about mathematical induction, a method of proof typically used to establish that a given statement is true for all natural numbers.

Easy Math Editor

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Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

The good part is "We now sit back and drink a cup of coffee" that really nice sir :)

Dhelia Dhelia - 8 years ago

At the end of the first worked example:

{We now sit back and drink a cup of coffee.]

You're mixing up your brackets. :)

Tim Vermeulen - 8 years ago

Sir, can you tell me, that how many types of induction are there?

Siddharth Kumar - 8 years ago

The works 'type' could either refer to various types of applications (first paragraph), or various mathematical topics where induction can be applied (second paragraph).

I have broadly identified 5-6 different applications that are based off the underlying idea of Induction, and will plan to explore them in the upcoming weeks. For example, students generally have heard of the Non-standard Induction, where you use several base cases instead of just 1, or the Strong Induction, where you use every preceding statement, instead of just $P_k$ .

Almost every mathematical topic has a problem which yields to an inductive approach. In the post, I've presented a question on summation of series, and another on divisibility. Other areas include recurrence relations, inequalities, functional equations, integration/differentiation, games, etc.

Edit: Here are various 'types' of induction. This might not be a complete list.

The first principle of finite induction is mentioned in this post.
The second principle of finite induction is also known as strong induction.
The third principle of finite induction is also known as non-standard induction.
There's also Forward-Backward Induction, as used in the Proof of AM-GM.
I've referred to something that I call 'stronger induction', which means proving a stronger statement than what was given.
There's the principle of transfinite induction, which extends it from the integers to well-ordered sets like the ordinals or the cardinal numbers. This requires the axiom of choice.
Structural induction, used in logic, computer science, graph theory,
Well-founded induction, also known as Noetherian induction.

Calvin Lin Staff - 8 years ago

there are three types of induction, first principle induction second principle induction third principle induction. if u want to know about them, just reply here.

Shobhit gautam - 8 years ago

@Shobhit Gautam – please tell

superman son - 8 years ago

@Shobhit Gautam – yes, please tell about them.

Siddharth Kumar - 8 years ago

@Shobhit Gautam – I'm fairly certain that there are only two principles; I have never heard of a third.

The first principle is simple induction, and the second principle is the Strong Induction that Calvin alludes to in his comment.

David Altizio - 8 years ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$