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quentinchin
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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).
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
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.
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.
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.
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.
Zurtex said:I'm curious, what level of maths do you do?
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?
The statement being asked to prove is the divisibility of 32n+1 + 2n+2 by 7.
In this statement, "divisibility" means that the expression 32n+1 + 2n+2 is evenly divisible by 7, meaning that there is no remainder when dividing the expression by 7.
This statement is being proved in order to show that there is a specific pattern in the results when dividing powers of 3 and 2 by 7, and to help understand the concept of divisibility and its applications in mathematics.
The exponents (2n+1) and (n+2) represent the powers of 3 and 2, respectively. These exponents are used in the statement to show the specific pattern and relationship between the two numbers and their divisibility by 7.
An expert can use their knowledge and understanding of mathematical concepts, such as divisibility rules and properties of exponents, to provide a logical and step-by-step explanation of the proof for the statement. They can also provide examples and counterexamples to further illustrate the concept and answer any specific questions or concerns about the statement.