anonymous
  • anonymous
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
Mathematics
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anonymous
  • anonymous
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
Mathematics
schrodinger
  • schrodinger
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amistre64
  • amistre64
you have the rule, and the starting values ... just use them
anonymous
  • anonymous
If I knew how to do that I probably would have done that. I have no idea how to do this problem.
amistre64
  • amistre64
let n=2, what does the rule become?

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amistre64
  • amistre64
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 ...
amistre64
  • amistre64
then let n=3, then 4, then 5 and you will generate the list of values with each new calculation
anonymous
  • anonymous
Oh, I see. I have to write out each one
amistre64
  • amistre64
it would help yes .... the process is short enough that it is the most efficient method
amistre64
  • amistre64
if youhad to find say a103 then finding a closed form would be more suitable
amistre64
  • amistre64
or writing a computer code to work it thru for you :)
anonymous
  • anonymous
I still do not think I am doing this correctly. My answer is x+2x^2+5x^3+13x^4+34x^5
amistre64
  • amistre64
a_(2) = 3 a_(1) - a_(0) but a_(0) = 1 and a_(1) = 1 so, a_(2) = 3(1) - 1 = 2
amistre64
  • amistre64
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 ...
anonymous
  • anonymous
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anonymous
  • anonymous
Does that mean my generating function is correct?

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