What is the easiest mathematical induction inequality prof method? (1 Viewer)

[barbernator]

I have seen quite a few different methods, what do u guys think is the easiest?

 

[math man]

if you look at 4d of http://www.boardofstudies.nsw.edu.au/hsc_exams/hsc2005exams/pdf_doc/maths_ext1_05.pdf

we have to prove:



for

now step 1 is easy so i'll skip it and go to step 2.

Step 2 Assume true for n=k



Step 3 To prove true for n = k + 1 (Now this is the step that gets hard and there are many ways to do it)

Prove:



Now what you should also do is use the assume n=k to prove this, so that means we will need the above inequality, however first i will rearrange it as follows:



So now we know this is true so we will need this somewhere. Now what you do is you take the LHS of the n=k+1 and try and manipulate it to prove the inequality as follows:

what i'll do now is change this as follows

i do this because we already know something about from above so now i sub in what knowledge i know as follows:

we must here change it to LHS> because remember so subbing this in makes the LHS greater than that.

Now we expand and simply this to:



and we know this is true for all so we have now proved true for n=k+1, and step 4 well yeah you know how to do it.

Hope this helps

 

[hup]

if you look at 4d of http://www.boardofstudies.nsw.edu.au/hsc_exams/hsc2005exams/pdf_doc/maths_ext1_05.pdf

we have to prove:




for

now step 1 is easy so i'll skip it and go to step 2.

Step 2 Assume true for n=k



Step 3 To prove true for n = k + 1 (Now this is the step that gets hard and there are many ways to do it)

Prove:



Now what you should also do is use the assume n=k to prove this, so that means we will need the above inequality, however first i will rearrange it as follows:



So now we know this is true so we will need this somewhere. Now what you do is you take the LHS of the n=k+1 and try and manipulate it to prove the inequality as follows:

what i'll do now is change this as follows

i do this because we already know something about from above so now i sub in what knowledge i know as follows:

we must here change it to LHS> because remember so subbing this in makes the LHS greater than that.

Now we expand and simply this to:



and we know this is true for all so we have now proved true for n=k+1, and step 4 well yeah you know how to do it.

Hope this helps

assume



then you have to prove for n = k+1 that



so use the assumption



then you try and make everything look like what you are trying to prove



now you know



so



so for inequality inductions you should write down what you have to prove then usually you'll have a double inequality using the conditions for n/k which proves it

 

[D94]

Because it's "induction", you shouldn't be subbing in values as the proof. I think it should be done like this:



In the second last line, we can show by induction again that 14k - 4 is > 0, but within the domain, 14k - 4 is always > 0 from inspection.

 

[math man]

well for the division induction questions subbing the assumption is the best way to do it