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Math Help - Number sequences

  1. #1
    Senior Member Mukilab's Avatar
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    Number sequences

    Lets say I have two number sequences

    First: 2,7,12,17
    Nth term: 5n-3

    Second:-4,-1,2,5,8
    Nth term: 3n-7

    If I wanted to find out several numbers which exist in both sequences without doing it manually what would I do? A simultaneous equation?
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  2. #2
    Senior Member Dinkydoe's Avatar
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    Depends on what you mean. If you want to find numbers a_n,b_n such that a_n=b_n for the same number n. Then you can equate: 5n-3=3n-7 to find out that no such number exists.

    But you want to find a_n and another index m such that a_n=b_m

    We can do that with a modulo argument:

    Let a_n = 5n-3 then a_n consists of all natural numbers, a_n\equiv 2 \text{mod} 5

    Let b_m=3m-7 then we must find all m such that b_m\equiv 2 \text{mod} 5

    3m-7\equiv 2 \text{mod} 5 \Rightarrow m\equiv 3 \text{mod} 5

    Thus b_m is in the sequence (a_n)_n if and only if m=5k+3 for any k.

    Thus the sub-sequence  (b_{5n+3}) = (15n+2) is in (a_n), n\geq 0

    Or (15n-13), n\geq 1
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  3. #3
    MHF Contributor
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    Quote Originally Posted by Mukilab View Post
    Lets say I have two number sequences

    First: 2,7,12,17
    Nth term: 5n-3

    Second:-4,-1,2,5,8
    Nth term: 3n-7

    If I wanted to find out several numbers which exist in both sequences without doing it manually what would I do? A simultaneous equation?

    yes, that will work (you would have to be careful only to look for integer solutions to those equations).


    It might be easier to note that every positive integer that ands in "2" or "7" is in sequence A.

    So to find the common numbers, just find the terms in sequence B that end in "2" or "7" (ignore the negative ones at the start)


    edit Too slow...again
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  4. #4
    Senior Member Mukilab's Avatar
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    Thanks but I didn't really understand Dinkydoe's explanation.
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  5. #5
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    Do you know what the modulo function is?
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  6. #6
    Senior Member Mukilab's Avatar
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    No but I've used it before. I used it in a case of where I had 5 numbers on this circle and at each rotation it would go up a number and the mod functino calculated so that I had every number on every 4th rotation.

    Something to do with the enigma machine.
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