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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.

Guest Sep 5, 2018

#3**+1 **

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)**

Guest Sep 6, 2018

#4**0 **

I dont think that is right.

Can somebody help me on this again?

Guest Sep 6, 2018

edited by
Guest
Sep 6, 2018

#7**+1 **

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."

Guest Sep 7, 2018

#8**+1 **

I agree with your terms, but your answer, 1/20 (3* 4^n - 8* (-1)^n), is not correct.

Guest Sep 7, 2018

edited by
Guest
Sep 7, 2018

#9**+1 **

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))

Guest Sep 7, 2018

edited by
Guest
Sep 7, 2018

edited by Guest Sep 7, 2018

edited by Guest Sep 7, 2018