Proven: n^3 + 2n is Divisible by 3 for Any Natural Number n

In summary, the equa (left hand side) is equal to the equa (right hand side) which is to say that if n is any natural number, n^3+2n is divisible by 3.
  • #1
teng125
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0
anyone pls help...for this ques:For any natural number n, n^3 + 2n is divisible by 3.

i don't know how to start or do
 
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  • #2
What does induction tell you to do? If you know what induction is then you know how to do this, so start by writing out what you need to do for induction. if you don't understand that then the rest of the question is not important.
 
  • #3
first to prove that left hand side of the equa is equal to right hand side or the equa which i did it.
then the sec step which is it i should prove that (n+1)^3 + 2(n+1) is also divisible by 3??ah...this step i can't proof
 
  • #4
left hand side of what 'equa' is equal to the right hand side of what? Or what is the second 'equa'? (Heck, what was the first).

Show that n^3+2n is divisible by three if n=1 (or zero if you like), now show that if k^3+2k is divisible by three then so is (k+1)^3+2(k+1)

note if X is divisible by three and Y is divisible by three then so is X+Y. So what happens if you subtract k^3+2n from (k+1)^3+2(k+1)?
 
  • #5
You titled this "induction". Are you saying you do not know what "proof by induction" means? There are two steps to induction and you certainly should be able to do the first!

Do this: open your textbook and look up "proof by induction". Tell us precisely what you need to do to prove "n3+ n is divisible by 3" and we'll help you go from there.
 
  • #6
Ok. To do mathematical induction, I was taught to do this in 3 steps. These 3 steps are:

1) Show true for n = 1
2) Assume true for n=k.

By doing this assumption, you set up for the third step by proving true for n=k+1, which will prove that n = k is also true.

3) Prove true for n = k+1.

Follow these steps throughout and see how it goes from there. I'll give you a little head start but try to finish it off.

Step 1: Show true for n=1
[tex]1^3+2(1) = 3[/tex] (which is divisable by 3)

Step 2: Assume true for n = k
i.e. Assume [tex]3 | k^3+2(k)[/tex] (induction hypothesis)

Step 3: Prove true for n = k+1
i.e. Prove that [tex] 3 | (k+1)^3+2(K+1)[/tex]

Expand out [tex](k+1)^3+2(k+1)[/tex] and when you fully simplify it out, try to keep your induction hypothesis ([tex]k^3+2(k)[/tex]) separate and see what's left over in your simplified expression. It should result in something that is divisible by 3. Show us how you expand it out and if you make any errors, we'll help you out gladly. But if you don't do the work, we won't help you. It's as simple as that. Effort needs to be shown, not just a problem that's shoved in our faces.
 
Last edited:
  • #7
Sorry for double post. This forum really doesn't work well with firefox.
 
  • #8
oh...okok...i found the answer already.thank you for ur help
 

1. What does the statement "Proven: n^3 + 2n is Divisible by 3 for Any Natural Number n" mean?

This statement means that for any natural number, when it is raised to the third power and added to twice itself, the resulting number is divisible by 3. In other words, the remainder when dividing n^3 + 2n by 3 is always 0.

2. How was this statement proven?

This statement was proven using mathematical induction. The base case (n=1) was shown to be true, and then it was shown that if the statement holds for n, it also holds for n+1. Therefore, the statement holds for all natural numbers.

3. What is the significance of this statement in mathematics?

This statement is significant because it is a fundamental property of natural numbers and can be used in various mathematical proofs and applications. It also demonstrates the power of mathematical induction as a proof technique.

4. Can this statement be extended to other numbers besides natural numbers?

No, this statement only holds for natural numbers. If n is a negative number, n^3 + 2n will not be divisible by 3. Additionally, if n is a non-integer, the statement does not hold.

5. Is there a practical application of this statement?

While this statement may not have a direct practical application, it is a useful property in number theory and can be used to prove other theorems and solve problems in mathematics. It also serves as a basis for understanding divisibility rules and patterns in numbers.

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