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Cutiepo0

  • 3 years ago

Write an explicit formula for the sequence determined by the recursion formula \[t n = 0.5t _{n-1}+n\] t1=40, t2=22, t3=13, t4 = 8.5, t5=6.25 Please explain how to get to the answer rather than just the answer. Thanks :)

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  1. mahmit2012
    • 3 years ago
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    |dw:1339363260817:dw|

  2. asnaseer
    • 3 years ago
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    I used this site to help me understand how to solve this: http://hcmop.wordpress.com/2012/04/20/using-characteristic-equation-to-solve-general-linear-recurrence-relations/

  3. mahmit2012
    • 3 years ago
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    |dw:1339363321532:dw|

  4. mahmit2012
    • 3 years ago
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    your answers are not correct

  5. Cutiepo0
    • 3 years ago
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    yeah I just realized that

  6. asnaseer
    • 3 years ago
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    yes - the equation I ended up with agrees with the values found by mahmit2012

  7. asnaseer
    • 3 years ago
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    I can let you know what equation I found if you want - or do you want to try it yourself first?

  8. Cutiepo0
    • 3 years ago
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    I know the answer, it's tn= ((n+1)(n+2))/2 but I don't know how to get to it

  9. asnaseer
    • 3 years ago
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    I haven't done problems like this before but found that document and followed it to get the answer

  10. asnaseer
    • 3 years ago
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    I actually got:\[t_n=80(\frac{1}{2})^n+2(n-1)\]

  11. Cutiepo0
    • 3 years ago
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    yeah but the textbook gave me the answer I posted. And that website uses very complex explanations :$

  12. asnaseer
    • 3 years ago
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    your formula does not produce the right answers

  13. asnaseer
    • 3 years ago
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    take n=1 - it produces the value 3

  14. Cutiepo0
    • 3 years ago
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    yeah I just checked, you're right it gave me the wrong answer.

  15. asnaseer
    • 3 years ago
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    if you use the formula I derived then it matches all the values (listed by mahmit2012)

  16. asnaseer
    • 3 years ago
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    I can try and explain the steps I took to derive this

  17. Cutiepo0
    • 3 years ago
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    okay, thanks that would be good

  18. mahmit2012
    • 3 years ago
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    |dw:1339364532207:dw|

  19. asnaseer
    • 3 years ago
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    ok - first it said arrange the recurrence relation in the form:\[At_n+Bt_{n-1}+Ct_{n-2}+...=f(n)\]so I arranged your relation as:\[t_n-0.5t_{n-1}=n\tag{a}\]it then said the general solution when f(n) is zero is \(t_n=A(0.5)^n\). It then said that to take account of f(n) we need to add in a solution of the form \(Bn+C\). So we substitute this into (a) to get:\[Bn+c-0.5(B(n-1)+C)=n\]which leads to:\[0.5Bn+0.5(B+C)=n\]and if we equate coefficients of LHS and RHS we get:\[B=2\]\[B+C=0\implies C=-B=-2\]so the second part of the equation is of the form:\[2n-2\]we add this to the first part to get the answer:\[t_n=A(0.5)^n+2n-2\]then use the fact that \(t_1=40\) to find A.

  20. mahmit2012
    • 3 years ago
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    |dw:1339364595544:dw|

  21. mahmit2012
    • 3 years ago
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    |dw:1339364644950:dw|

  22. mahmit2012
    • 3 years ago
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    |dw:1339364681732:dw|

  23. mahmit2012
    • 3 years ago
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    |dw:1339364741953:dw|

  24. mahmit2012
    • 3 years ago
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    |dw:1339364774607:dw|

  25. asnaseer
    • 3 years ago
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    hope it all makes sense?

  26. Cutiepo0
    • 3 years ago
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    thanks a lot guys :)

  27. asnaseer
    • 3 years ago
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    yw :)

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