We use cookies to personalise content and advertisements and to analyse access to our website. Furthermore, our partners for online advertising receive pseudonymised information about your use of our website. cookie policy and privacy policy.
 
+0  
 
0
171
10
avatar

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.

 Sep 5, 2018
 #1
avatar
0

Sombody help me plz

 Sep 6, 2018
 #2
avatar
0

Why not help me?

 Sep 6, 2018
 #3
avatar
+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)

 Sep 6, 2018
 #4
avatar
0

I dont think that is right.

Can somebody help me on this again?

 Sep 6, 2018
edited by Guest  Sep 6, 2018
 #5
avatar
+1

Give us a hint. What do you think "a(2)" should be?

 Sep 6, 2018
 #6
avatar
+1

a(2)=10

 Sep 6, 2018
 #7
avatar
+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."

 Sep 7, 2018
 #8
avatar
+1

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

 Sep 7, 2018
edited by Guest  Sep 7, 2018
 #9
avatar
+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))

 Sep 7, 2018
edited by Guest  Sep 7, 2018
edited by Guest  Sep 7, 2018
 #10
avatar
+1

wait, i see. Thanks you fro your help! Your help is appreciated!!!

 Sep 7, 2018

13 Online Users

avatar
avatar