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Math Help - Given two arithmetic sequences...

  1. #1
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    Given two arithmetic sequences...

    Given the following 2 arithmetic sequences, determine the first ten terms common to both sequences.

    Sequence #1
    3, 14, 25, ...

    Sequence #2
    2, 9, 16, ...

    I figured out that the two general formulas would be tn = 3 + 11 (n-1) and tn = 2 + 7 (n-1), respectively... But I'm not sure where to go from there. I've asked for help elsewhere but I still don't get it, and my test is coming up soon... =/ Thanks a lot.
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  2. #2
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    Quote Originally Posted by galactus View Post
    They just want the first ten terms, just add 11 in the first and 7 in the second.

    The first one: 3,14,25,36,47,58,69,....... and so on
    Is there no other way to do it besides writing the entire sequences out, and then circling the ones in common?
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  3. #3
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    If I'm reading your original post correctly, you don't want the first ten terms of each sequence, right? You want the first ten terms that are COMMON to BOTH sequences. If so:

    Here's what I did - it's slightly better than listing out every term.

    The first sequence's terms can be described by 2 + 7x.

    The second sequence's terms can be described by 3 + 11y.

    I want to know when these are equal, so:

    2 + 7x = 3 + 11y.

    I solved for y and got:

    y = \frac{7}{11}x - \frac{1}{11}.

    So I'm looking for an integer value of x that gives me an integer value for 7 (since all the numbers in both sequences are integers). I found the first one by trial and error. x=8 gives y=5. So the point (8,5) is on my line, and gives 58 for both 2 + 7x and 3 + 11y. (Points in which both coordinates are integers are called "lattice points" - there may be an easier way to find them, but I don't know it. Essentially, that's what we're looking for here - lattice points on our line.)

    So 58 is in both sequences. Now, noting that the slope of our line is 7/11, we can increase our y value by 7 and our x value by 11 (since slope is change in y / change in x) to get the next value of x and y that gives integer coordinates: (19, 12). Plugging x = 19 and y = 12 into the appropriate equations gives 135 in both, so that's the next term.

    Keep using slope to "jump" from each integer-coordinate point to the next. Use x and y, plug them into your equations, and you should be good to go:

    (8,5), (19,12), (30, 19), (41, 26), ...

    Hope this makes sense. I kind of figured it out while writing this, so it's a little... disorganized.
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  4. #4
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    Quote Originally Posted by Mathnasium View Post
    If I'm reading your original post correctly, you don't want the first ten terms of each sequence, right? You want the first ten terms that are COMMON to BOTH sequences. If so:

    Here's what I did - it's slightly better than listing out every term.

    The first sequence's terms can be described by 2 + 7x.

    The second sequence's terms can be described by 3 + 11y.

    I want to know when these are equal, so:

    2 + 7x = 3 + 11y.

    I solved for y and got:

    y = \frac{7}{11}x - \frac{1}{11}.

    So I'm looking for an integer value of x that gives me an integer value for 7 (since all the numbers in both sequences are integers). I found the first one by trial and error. x=8 gives y=5. So the point (8,5) is on my line, and gives 58 for both 2 + 7x and 3 + 11y.

    So 58 is in both sequences. Now, noting that the slope of our line is 7/11, we can increase our y value by 7 and our x value by 11 (since slope is change in y / change in x) to get the next value of x and y that gives integer coordinates: (19, 12). Plugging x = 19 and y = 12 into the appropriate equations gives 135 in both, so that's the next term.

    Keep using slope to "jump" from each integer-coordinate point to the next. Use x and y, plug them into your equations, and you should be good to go:

    (8,5), (19,12), (30, 19), (41, 26), ...

    Hope this makes sense. I kind of figured it out while writing this, so it's a little... disorganized.
    Could you not use the variation of the Euclidean algorithim on this since it is a Diophantine equation?

    Or am I stepping out of my bounds here?
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  5. #5
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    The answer that I posted before was correct, BUT the notation was mistaken.
    Allow me to change the notation to an index of zero:
    a_0  = 2,\,a_1  = 9,\, \cdots ,\,a_n  = 2 + 7n & b_0  = 3,\,b_1  = 14,\, \cdots ,\,b_n  = 3 + 11n.

    Now it is easy to check that: a_8  = b_5 ,\,a_{19}  = b_{12} ,\,a_{30}  = b_{19} ,\, \cdots .

    Do you see the relation among the subscripts: 8,19,30 & 5,12,19?
    Are they  - 3 + 11j,\,j = 1,2,3, \cdots and  - 2 + 7k,\,k = 1,2,3, \cdots ?
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