Consider the sequence defined by
\(\begin{cases} a_0=1\\ a_1=2\\ a_n=3a_{n-1}+4a_{n-2} & \text{if }n\ge 2 \end{cases}\)
Find a closed form for \(a_n\).
Your response should be a formula in terms of n.
If I understand your sequence, you should have the following:
a(0) = 1
a(1) = 2
a(2) = 3*2+4*1 =10
a(3) =3*10 + 4*2= 38
a(4) =3*38 + 4*10 =154
a(5) =3*154 + 4*38=614
a(6) =3*614 + 4*154 =2,458
a(7) =3*2,458 + 4*614 =9,830
a(8) =3*9,830 + 4*2,458 =39,322
So, the sequence should continue as follows:
1, 2, 10, 38, 154, 614, 2458, 9830, 39322, 157286, 629146, 2516582, 10066330, 40265318, ...etc.
The closed form would then be:a_n = 1/20 (3* 4^n - 8* (-1)^n) (for all terms given)
I dont think that is right.
Can somebody help me on this again?
So, if a(2) =10, then a(3) =3(a3 - 1) + 4(a3 - 2) =3*(a2) + 4*(a1)=3*10 + 4*2 =38. You don't agree with this? What should a(3) be? I would like to help you if I can. If we can agree on the terms of the sequence, then we can find the "closed form."
I agree with your terms, but your answer, 1/20 (3* 4^n - 8* (-1)^n), is not correct.
Try the 8th term that I have calculated above:
8th term =:a_n = 1/20 (3* 4^n - 8* (-1)^n) (for all terms given)
a_9 = 1/20(3* 4^9 - 8*(-1)^9
=1/20( 786,432 - 8*-1)
=1/20(786,432 + 8)
=1/20(786,440)
=39,322
As you can see, the a(8) term, which is the 9th term, calculates it accurately. The closed form that you may have may be easier or simpler, but essentially is equivalent to this one. If you know a simpler "closed form" formula then used it instead of this one. Mine calculates EVERY term given accurately. You may test it on any term you wish.
Note:After a(2) term, you may use this "closed form" formula:a_n = 2/5 ((-1)^n + 3 2^(2 n + 3))
