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QCE Specialist Maths Questions Thread
lamh21:
--- Quote from: keltingmeith on October 07, 2020, 12:23:56 pm ---Yeah, so the trick with these types of induction proofs is you want to show that the LHS turns into some RHS that's equal to 240*(some number). For example, 480 is a multiple of 240 because 480=240*(2). If you were to get something disgusting out the end that looks like n^5-n=240*(n^100 - n^432 + 54432323n^2 + 5 - 45n) or something else that stupid, it's still 240*(some number), and so is still divisible by 240. Since it's an odd number, you also need to make sure your inductive proof only goes through odd numbers. There are two ways to do this:
1. Prove this is true for n=2m+1, instead of for n, or:
2. Make sure your base case is an odd number (so, start with n=1), then prove it true for n=k+2 instead of n=k+1 (WHY would this work??)
Otherwise, it's still an inductive proof at its heart. I want you to try this out for yourself first, so here's an example using an example question that you can use to see my hints in action:
Prove by induction that n3-n is divisible by 24 for all odd positive integers
So, first, the base case - n=1:
Which is divisible by 24. So, step 2 - assume it's true for n=k.
... Done
Okay, step 3. Let's see if this is true for n=k+2:
Okay, so we know that k^3-k is divisible by 24, so I'm going to substitute a 24x into there - because we don't care WHAT value it is exactly, just that it IS divisible by 24. I also know that k is an odd number, so k+1 HAS to be even - so I'm just going to call that 2y. Because again, I don't care EXACTLY what the number is, I just care about what it's divisible by. So this gives me:
Which is a multiple of 24, and completes the proof
---
So, some questions I often get asked:
a) how did I know to make that expansion and factorisation in the steps?
I didn't - but it was either do that or do anything. With all of these proofs, I have no idea what direction I need to move. But, if I don't move forwards, I won't get anywhere - expanding at the start is the only thing I could do, so I did it. And every time I expand something, I expect I need to factorise it later, so when I recognise something I CAN factorise - I do it. If I factorise, and it turns out that that's NOT useful, then I can always just expand it in the next step and move on.
b) how did I know to make k+1=2y?
I didn't. All I know is, the more I can reduce things to stuff they're divisible by, the easier these proofs become - so I saw I could turn k+1 into a multiple of something, and I ran from there.
---
Also, if you're interested, here's how you'd do it using the set n=2m+1 method.
Step 1: Prove this is true for the base case, m=0:
Done, simple. Now, assume this is true for m=k
... Done
Now, let's test m=k+1:
From here, just pick another variable (say, L=2k+1), and this follows the same steps as the one above.
--- End quote ---
Ahh I see! Thank you so much for your help! :)
Adfer:
Hi, I’ve attempted the attached question and I’m quite stuck. I have also attached my attempted solution, however I am unsure where I have gone wrong. Thanks in advance for any help :)
Bri MT:
--- Quote from: Adfer on October 09, 2020, 05:22:22 pm ---Hi, I’ve attempted the attached question and I’m quite stuck. I have also attached my attempted solution, however I am unsure where I have gone wrong. Thanks in advance for any help :)
--- End quote ---
Hi,
Welcome to the forums!
I can't see any attached question, could you please edit your post and try attempting to attach it again, embedding an image of the question, or writing the question out for us? if you're unsure about how to do this there are instructions here and please feel free to ask as well :)
Adfer:
--- Quote from: Bri MT on October 09, 2020, 05:28:10 pm ---Hi,
Welcome to the forums!
I can't see any attached question, could you please edit your post and try attempting to attach it again, embedding an image of the question, or writing the question out for us? if you're unsure about how to do this there are instructions here and please feel free to ask as well :)
--- End quote ---
Haha my apologies, took a while to attach everything - but it should be there now. I've attached the final part of my working to this post as it was too large to put all on the post above.
Britnium:
Hello!
I'm currently doing a PSMT on Leslie Matrices; I had believed that I've gotten a good chunk of the assignment done but after hearing other information floating around the grade I think I'm a little lost.
So our task is to model population trends of the Tasmanian devil since the documentation of the Devil Facial Tumor Disease in 1996 up until 2030 to determine whether or not the species will go extinct.
I've calculated the initial female age distributions and now I just need to develop a Leslie matrix (which is 7x7) to model the population trends. Currently, they have provided us with the following birth and survival rates, all of which are for healthy devils, so the challenge I'm facing is determining these rates for disease-affected populations.
Survival rates for rates based on historical data for disease-free populations (where s0 = probability of surviving the 0-1 age interval):
s0 = 0.39
s1 = 0.82
s2 = 0.82
s3 = 0.82
s4 = 0.82
s5 = 0.27
s6 = 0
Breeding numbers (female per female devil):
m0 = 0
m1 = 0.03
m2 = 0.86
m3 = 1.55
m4 = 1.55
m5 = 1.55
m6 - 0.86
We've also been provided with relevant research, which I'm 99% sure we're to use for developing our survival rates. Please see the screenshots attached.
My initial guess was that we were to select appropriate survival rates from the range of 0.1-0.6, model the trends using these numbers and compare the obtained populations with actual statistics (e.g. 50% killed from 1996-2007) to establish validity in the model and change the rates where necessary to match up with these figures and thus 'refine' our model. However, recently there seems to be a stress on the sentence 'a large host population will experience a rapid decline followed by stabilization and eventual return to pre-disease numbers.' Under the assumption that birth rates will stay the same, I'm completely unsure of how to obtain survival rates that would give us this stabilization point and subsequently model the return to pre-disease numbers. And would this all be done under one Leslie matrix? Or would we expect to have multiple to model different periods of time?
Any help on this would be greatly appreciated. Sorry for such the long question and apologies if my stress has gotten to you too :'(
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