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anonymous

  • one year ago

Let A(x)= SUM(a_nX^n) be the generating function of the sequence a_0, a_1, a_2, ... that is recursively defined by a_0=a_1=1 and a_n=3(a_(n-1))-(a_(n-2) where (n>=2). Compute a_5

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  1. amistre64
    • one year ago
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    you have the rule, and the starting values ... just use them

  2. anonymous
    • one year ago
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    If I knew how to do that I probably would have done that. I have no idea how to do this problem.

  3. amistre64
    • one year ago
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    let n=2, what does the rule become?

  4. amistre64
    • one year ago
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    a_n = 3 a_(n-1) - a_(n-2) 2 2 2 a_2 = 3 a_(2-1) - a_(2-2) a_2 = 3 a_(1) - a_(0) and we have the values for 0 and 1 already stated ...

  5. amistre64
    • one year ago
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    then let n=3, then 4, then 5 and you will generate the list of values with each new calculation

  6. anonymous
    • one year ago
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    Oh, I see. I have to write out each one

  7. amistre64
    • one year ago
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    it would help yes .... the process is short enough that it is the most efficient method

  8. amistre64
    • one year ago
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    if youhad to find say a103 then finding a closed form would be more suitable

  9. amistre64
    • one year ago
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    or writing a computer code to work it thru for you :)

  10. anonymous
    • one year ago
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    I still do not think I am doing this correctly. My answer is x+2x^2+5x^3+13x^4+34x^5

  11. amistre64
    • one year ago
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    a_(2) = 3 a_(1) - a_(0) but a_(0) = 1 and a_(1) = 1 so, a_(2) = 3(1) - 1 = 2

  12. amistre64
    • one year ago
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    lets forgo the _(n) stuff becuase its a bugger to type a3 = 3 a2 - a1, but we know a2 and a1 a3 = 3(2) - 1 = 5 -------------------- a4 = 3 a3 - a2, but we know a3 and a2 a4 = 3(5) - 2 = 13 --------------------- a5 = 3 a4 - a3, but we know a4 and a3 ...

  13. anonymous
    • one year ago
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  14. anonymous
    • one year ago
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    Does that mean my generating function is correct?

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