Question

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

Answer 1

you have the rule, and the starting values ... just use them

Answer 2

If I knew how to do that I probably would have done that. I have no idea how to do this problem.

Answer 3

let n=2, what does the rule become?

Answer 4

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 ...

Answer 5

then let n=3, then 4, then 5

and you will generate the list of values with each new calculation

Answer 6

Oh, I see. I have to write out each one

Answer 7

it would help yes .... the process is short enough that it is the most efficient method

Answer 8

if youhad to find say a103 then finding a closed form would be more suitable

Answer 9

or writing a computer code to work it thru for you :)

Answer 10

I still do not think I am doing this correctly. My answer is x+2x^2+5x^3+13x^4+34x^5

Answer 11

a_(2) = 3 a_(1) - a_(0)

but a_(0) = 1 and a_(1) = 1

so, a_(2) = 3(1) - 1 = 2

Answer 12

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
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a5 = 3 a4 - a3, but we know a4 and a3 ...

Answer 13

Does that mean my generating function is correct?