Mathematical Induction - Divisibility Tests (1) | ExamSolutions
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- เผยแพร่เมื่อ 5 ต.ค. 2024
- Here I look at using proof by mathematical induction for divisibility tests.
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Last hour prep (literally had been crying) and this man is here to give you the best in a few minutes
I've been working with this thing for hours, and now with the help of your video I could finally solve it
You've just saved a life.
Thank you for this really understandable video :)
Thank you, that would be really appreciated.
Wow thank you so much
I've been finding it so difficult to solve questions like this
You really help me a lot
Keep it up!
Thank you so so much!!!
How does he get to the conclusion that it is 8(3^2k) - my question goes as to how it is 8, in particular, and what happens to the - 3^2k? Cheers guys, I'm a total rookie who just need to understand how the algebraic expression is simplified?!?
I'm confused as well
3^2k(3^2-1)
3^2K(8)
If I had found this video earlier in my study, I could save 30mins.
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Can you please explain why we don't add the n= k+1 to n=k? I am a little confused on the part where you did f(k+1)-f(k) and why you did that.
I understand the rest but if you can just explain that part then all the gaps will be filled :) .
Thanks
Did you watch all the video? At the beginning I said this (and this applies to all divisibility tests) was handled differently from many of the other forms of proof by induction by this method of doing f(k+1)-f(k). We show that f(k+1)-f(k) = some expression and then f(k+1)=the expression +f(k). All we need to show is the expression is divisible by 8 and we know that f(k) is divisible by 8 so if both terms are divisible by 8 then f(k+1) must be divisible by 8.
@@ExamSolutions_Maths You are not answering , You are just telling what You did
cool.
Thankyou very much for making me understand this concept
Nice teaching
Thank you and pleased to hear it helped.
Very good!
Many thanks.
how'd you get 8 in f(k+1)
in 5:02 , If you expand the 3^2 in the front, you will get 9(3^2k) - 3^2k which equals to 8(3^2k)
how did u get 3squre{3^2K}-3^2K
amazing explanation
great method. thanks. would this method work for something such as this....... prove 2^(3n) - 3^n is divisible by 5 for all natural numbers
Yes it should
@@ExamSolutions_Maths iv gotten as far as f(k+1) - f(k)= 7(2^3k) -2(3^k) ...........but how do i know this is also divisible by 5. thanks in advance
Sorry for the delay in replying but a lot has been going on over the last few days. You may have it sorted by now but this is tricky.
Now we know that f(k)=2^3k - 3^k and it is divisible by 5 (assumption) so rearranging 2^3k = f(k) + 3^k
Sub this into your last statement f(k+1) - f(k)= 7(2^3k) -2(3^k) to give
f(k+1) - f(k)= 7[f(k) + 3^k] -2(3^k)
so
f(k+1) - f(k)= 7 f(k) + 7(3^k) -2(3^k)
which gives
f(k+1) - f(k) = 7f(k) + 5(3^k)
Now add f(k) to both sides
so
f(k+1)=8f(k) + 5(3^k)
Now f(k) is assumed to be divisible by 5 so f(k)=5a where a is a positive integer
so
f(k+1)=40a + 5(3^k)
so
f(k+1)=5[8a +3^k]
Which is a multiple of 5 so hence divisible by 5 as 8a+5(3^k) is an integer
I hope that answers your question.
@@ExamSolutions_Maths after 3 days of messing around with it i finally got it sorted. thanks for your reply though. great videos and a very nice way of doing the divisibility proofs.
Can anyone tell me how 3^2.3^2k - 3^2k = 8(3^2k)???
Please guide me about that if anyone is reading my comment, I'm stuck here 🥺🥺
Okay first lets simplify 3^2 which is 9 so we have: 9(3^2k)-3^2k
Next we rewrite the statement like this: 9(3^2k) + (-1x 3^2k) which is the same as 9(3^2k)-3^2k we just multiplied -1 times 3^2k which is the same as subtracting it
Now we can factor out 3^2k like this: 3^2k(9-1)
The last step is to calculate 9-1 so we get 3^2k(8) which is the same as 8(3^2k)
I hope that helps took me a while to figure this out
@@ketchupjunge2826Danko
Thank you!
You're welcome Matti! All the best
how would you use proof by induction to prove a function is not divisible by a real number, like n^3 +2 for example is not divisible by 8
uhh what year group is this for...
Are you using a Blue Yeti microphone?
Yes
dankie
Marvelous
is this pre-algebrq, algebra or calculus?
algebra I guess
I miss quadratic equation...
why did you add f(k) I do the same method but I dont add f(k) and i make F(k+1) = 8A and then i Factor
I added f(k) so that f(k+1) was left on the LHS and on the RHS I had two term which contained a factor of 8. Remember that I assumed that f(k) was divisible by 8.
Show me the method you used and I can then see if it is a fair argument.
mathematical fking
TI SI BOG
Where does that 8 come from cause 3² is 9