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Which cube members are not in the sequence and prove it?

  1. Feb 12, 2006 #1
    Which cube members are not in the sequence and prove it?

    2, 5, 8, 11, 14, ...

    How can this be proved :cry:

    My answer:

    an = 3n + 2

    Any natural number may be written as N= 3k+p for some natural number K and p=0,1 or 2.

    So

    N^3=(3k+p)3
    N^3=3(9k+k^2p+kp^2)+p^3
    N^3=3k+p^3

    Therefore N^3 (mod 3)

    So as an is congruent to 2(mod 3), N^3 lies in the sequence. In other words, the cude of any number in the sequence is in the sequence. But how do I go and show which cube members are not in the sequence and moreover how do I prove it?
     
  2. jcsd
  3. Feb 12, 2006 #2

    AKG

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    You want to prove something of the form:

    a cube x³ is not in the sequence if and only if ________

    where you fill in the blank with something informative. You have to understand that about the question, that it is somewhat open-ended. Technically, you could fill in the blank with "x³ is not in the sequence" or "x³ is not congruent to 2 (mod 3)" but neither answer is very informative. I'm actually pretty stumped as to how to explain how to do this. If you start with the condition "x³ is not in the sequence," will you be able to churn out equivalent conditions such that you can prove:

    "x³ is not in the sequence" if and only if A
    A iff B
    B iff C
    C iff D

    Until you reach something (in this case, D) which is informative, making your final answer "x³ is not in the sequence iff D". Or should you guess the answer, D, first, then work your way to finding sentences A, B, and C such that you can prove:

    "x³ is not in the seq." iff A
    A iff B
    B iff C
    C iff D?

    Well, here are a few logical tautologies which you should understand (they hold for any sentences A, B, and C):

    "A iff B" is equivalent to "A implies B and B implies A"
    "A implies B" is equivalent to "not-B implies not-A"
    "A iff B" is equivalent to "A implies B and not-A implies not-B" (putting the above two together)
    "A iff B" is equivalent to "either both A and B are true, or both A and B are false - that is, A and B have the same truth-value, whatever that truth value may be"
    A iff A
    "A iff B, and B iff C" implies "A iff C", however
    "A iff C" does not imply "A iff B and B iff C", so
    "A iff C" is not equivalent to "A iff B, and B iff C"
    "A iff B" is equivalent to "B iff A"
    "A implies B" is NOT equivalent to "B implies A"
    "A iff B" is equivalent to "not-A iff not-B"
     
  4. Feb 12, 2006 #3

    AKG

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    So here are some good things to know:

    "x³ is not in the sequence iff A" is equivalent to "x³ IS in the sequence iff NOT-A"

    And

    "any number n is in the sequence iff n = 2 (mod 3)"

    So, in particular

    "a cube, n = x³ is in the sequence iff x³ = 2 (mod 3)"
     
  5. Feb 12, 2006 #4

    Physics Monkey

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    Natasha,

    Here is a hint to get you going: it is helpful to write your candiate number as [tex] N = 3k + r [/tex] where [tex] r = 0, \, 1, \, 2 [/tex]. You also correctly realized that when looking at [tex] N^3 [/tex], all the factors containing [tex] k [/tex] are already divisible by 3. Now, what you want is to have [tex] N^3 = 3 n + 2 [/tex] for some [tex] n [/tex], right? What does this say about the one factor in [tex] N^3 [/tex] that isn't explicitly divisible by [tex]3[/tex]?
     
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