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hey guy help me to prove this question:
3^(2n+1) + 2^(n+2) is dividable by 7.
3^(2n+1) + 2^(n+2) is dividable by 7.
Yes (assuming it's true), but this forum isn't about doing your work for you. It's just about helping.quentinchin said:Can u prove it ?
I'm a bit dubious about the logical jump there.LittleWolf said:3^(2n+1)+2^(n+2)=3*9^n+4*2^n. To be divisible by 7, you must prove
(9^n mod 7) =(2^n mod 7).
It's fine, you could then collect the common 2^n terms working mod 7, etc., but if you're already familiar with modular aritmetic, there shouldn't be much to prove.Zurtex said:I'm a bit dubious about the logical jump there.
Well whether or not it's fine, as you mention it would appear that quentinchin doesn't appear too familiar with modular arithmetic. Which is why I suggested just a simple proof by induction at the start of the thread.shmoe said:It's fine, you could then collect the common 2^n terms working mod 7, etc., but if you're already familiar with modular aritmetic, there shouldn't be much to prove.
It appears the quentinchin isn't familiar with modular arithmetic yet, so I offer the following hint: 9^n=(7+2)^n
Which he seemed to ignore (sigh) so I offered another suggestion. Here's hoping he actually gives either way a try and posts his efforts.Zurtex said:Which is why I suggested just a simple proof by induction at the start of the thread.
It may be because I'm very tired, but I don't see how you made that step.quentinchin said:= 3(9^n) - 3(2^n) + 7(2^n)
= 3(9-2)[9^(n-1) + 9^(n-2)*2 + ... + 9*2^(n-2) + 2^(n-1)] + 7(2^n)
hence the equation 3^(2n+1) + 2^(n+2) is dividable by 7.
I really appreciate it, but do you guys overlook the question? The question is to <PROVE> that 3^(2n+1) + 2^(n+2) is divisible by 7 but not using induction method.Zurtex said:It may be because I'm very tired, but I don't see how you made that step.
Anyway, proof by induction. First of all you prove it for n = 1, then assuming it's true for n = k you prove it for n = k + 1. That proves it for all natural numbers.
Here is a nice page on it: http://en.wikipedia.org/wiki/Mathematical_induction
Really good for this exact sort of problem.
No, proof by induction here is clearly the easiest way to prove it. Even if you use some other method to reduce the problem, induction is probably the easiest way to prove what you have.quentinchin said:I really appreciate it, but do you guys overlook the question? The question is to <PROVE> that 3^(2n+1) + 2^(n+2) is divisible by 7 but not using induction method.
Your orignial question put no restrictions on what techniques could be used. Besides, you've solved it already so what's the problem?quentinchin said:I really appreciate it, but do you guys overlook the question? The question is to <PROVE> that 3^(2n+1) + 2^(n+2) is divisible by 7 but not using induction method.
I really do not know how to solve this question and I just get the answer from my friend after I asked this question.shmoe said:Your orignial question put no restrictions on what techniques could be used. Besides, you've solved it already so what's the problem?
(By the way, my hint had nothing to do with induction, but the binomial theorem.)
Seems reasonable but I certainly would not want to do that for every time I got asked to prove:quentinchin said:Maybe mathematic induction is the best way to prove this sort of question but if we too often using this kind of method, our mathematic level wouldn't go any further.
I really do not know how to solve this question and I just get the answer from my friend after I asked this question.
Is that
3*9^n + 4*2^n
=3(7+2)^n + 4*2^n
=3[7^n + n*7^(n-1)*2 + ... + n*7*2^(n-1) + 2^n] + 4*2^n
=3[7^n + n*7^(n-1)*2 + ... + n*7*2^(n-1)] + 7*2^n
=7{3[7^(n-1) + n*7^(n-2) + ... + n*2^(n-1)] + 2^n}
this is also another good method.
I know that my level just pre-U, but in my point of view about math is, it's doesn't matter how much knowledge or formula I know but what the most important thing is I must try to apply what I gained in surrounding of me. Just like mathematic induction I will also using it to solve other question.Zurtex said:I'm curious, what level of maths do you do?
i totally agree with you.steven187 said:if you want to build your mathematical mind you should try to challenge your self to prove something in as many ways as possible weither simple or hard, coming from a person who wants to be a mathematician would i be wrong to say such things?