## 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 :)

1. mahmit2012

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2. asnaseer

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

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4. mahmit2012

5. Cutiepo0

yeah I just realized that

6. asnaseer

yes - the equation I ended up with agrees with the values found by mahmit2012

7. asnaseer

I can let you know what equation I found if you want - or do you want to try it yourself first?

8. Cutiepo0

I know the answer, it's tn= ((n+1)(n+2))/2 but I don't know how to get to it

9. asnaseer

I haven't done problems like this before but found that document and followed it to get the answer

10. asnaseer

I actually got:$t_n=80(\frac{1}{2})^n+2(n-1)$

11. Cutiepo0

yeah but the textbook gave me the answer I posted. And that website uses very complex explanations :\$

12. asnaseer

13. asnaseer

take n=1 - it produces the value 3

14. Cutiepo0

yeah I just checked, you're right it gave me the wrong answer.

15. asnaseer

if you use the formula I derived then it matches all the values (listed by mahmit2012)

16. asnaseer

I can try and explain the steps I took to derive this

17. Cutiepo0

okay, thanks that would be good

18. mahmit2012

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19. asnaseer

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

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21. mahmit2012

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22. mahmit2012

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23. mahmit2012

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24. mahmit2012

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25. asnaseer

hope it all makes sense?

26. Cutiepo0

thanks a lot guys :)

27. asnaseer

yw :)