Conclusion to Mathematical Induction? (1 Viewer)

[Tsylana]

Is there a correct, short generic way to conclude MI...? It seems my teachers are always picky on what you write as a conclusion in ur mi statements... ><"... whats a satisfying enough conclusion that most teachers will accept?

 

[gurmies]

I say, once i've proved true for n = k + 1:

Hence, if the formula is true for n = k, it is true for n = k + 1.

BUT the formula is true for n = 1

Therefore it is true for n = 2

and so on...


Thus, it is true for all integral n..

 

[lolokay]

as it is true for n=1 (or whatever other number) and if true for n=k it is true for n=k+1, it is true for all integers n>=1

 

[Aplus]

This is pretty generic. Just edit as necessary.

Hence, if the result is true for n = k, then it must be true for n = k + 1. Since it is true for n = 1, then it must be true for n = k + 1 = 2. Thus it is true for n = 2, n = K +1 = 3, and so on. Therefore, it must be true for all positive values of n.

 

[cutemouse]

Thus, it is true for all integral n..

integral values of n.

'integral' is an adjective

 

[addikaye03]

This is pretty generic. Just edit as necessary.

Hence, if the result is true for n = k, then it must be true for n = k + 1. Since it is true for n = 1, then it must be true for n = k + 1 = 2. Thus it is true for n = 2, n = K +1 = 3, and so on. Therefore, it must be true for all positive values of n.

Derek Bucanhan believes that iterative form is misleading, he says lolokay's method is correct. Im just saying it as he says it, i have no opinion on whether it is right or not. i used lolokay's.

 

[Just.Snaz]

This is pretty generic. Just edit as necessary.

Hence, if the result is true for n = k, then it must be true for n = k + 1. Since it is true for n = 1, then it must be true for n = k + 1 = 2. Thus it is true for n = 2, n = K +1 = 3, and so on. Therefore, it must be true for all positive values of n.

I used something similar.

If the result is true for n = k then it also is true for n = k+1. since it is true for n = 1 then it is also true for n = 2, 3, etc. therefore, it is true for integers n>=1

 

[Trebla]

The whole point of the long version of the conclusion is to help you understand what induction is all about.

If you've proved a formula is true for n = k + 1 using the formula you've assumed for n = k, then the formula is only true for n = k + 1 only if the formula is true for n = k. Since you've proved (without assumption) it is true for n = 1 (or whatever the initial value is), then its true for the next value n = 2. Since it is true for n = 2, it is then true for n = 3 etc...it's a bit like a domino effect. If the initial number is true, then it is true for everything after it.

Most teachers encourage the long version. However, in the HSC you can get away with writing the shorter version "Hence by induction it is true for all integers ..."
I personally just add a bit more just to make the connection clearer.
"The statement is true for n = k + 1, if it is true for n = k. Since it is true for n = 1, it is true for all positive integers n"
which is just the same as what lolokay gave...

 

[untouchablecuz]

My teacher says this is fine:

"It follows from parts A and B by the principle of mathematical induction, that the statement is true for all positive integers n > [something]."

Part A is the test of the starting value and part B is the n=k and n=k+1 part.

Is that all we need to get full marks in HSC for the conclusion?

 

[madsam]

Our teacher gets us to prove for n=1, n=2, and n=3, but he told us for exams and assessments we need only do n=1

Anyway, our conclusion is:

As it is true for n = 1, and n = k+1, it must be true for n = 1+1 = 2, and n = 2+1 = 3, and so on for all values of n

 

[bored of sc]

When you think about it Induction is cheating in a way (using an assumption).

 

[youngminii]

When you think about it Induction is cheating in a way (using an assumption).

I reckon Induction is stupid
But since it's in the Syllabible..

 

[Trebla]

When you think about it Induction is cheating in a way (using an assumption).

Yes, but you then relinquish the assumption since you've proved your initial case, because you only use the assumption for n = k + 1. Once you've proved it is true for n = 1 and n = k +1 holds only if n = k holds then you no longer need to use the assumption to back up the formula, because the initial case is true without assumption.
Induction is merely a generalisation of just subbing in numbers manually. For example, if you want to check if the formula: 1² + 2² + 3² + ....... + n² = n(n + 1)(2n + 1)/6, you'd have to sub in n = 1, n = 2, n = 3 etc to verify it is true. To avoid subbing in all numbers manually, you use induction.

Induction is common technique of proof used in university level mathematics as well because it is an easy way to prove many complicated formulae, which is why it is part of the syllabus.

 

[Tsylana]

lol has anyone ever thought about how people make up inductions... they must be pretty retardedly bored. xD.

 

[yorkstanham]

pretty sure just "therefore proved by induction" is sufficient