Go Premium for a chance to win a PS4. Enter to Win

x
?
Solved

Simplifying Linear Second Order Recurrence sequences...

Posted on 2011-03-13
8
Medium Priority
?
510 Views
Last Modified: 2012-05-11
I'm not sure how to write the correct subscript notation for this expression, but I've got a sequence

Un = (some expression)

and need to prove an identity, where the LHS has Un+1 Un-1 - (Un)^2

(where n, n+1 and n-1 are all subscripts of U)

By multiplying out the expression substituting in n, n-1 and n+1 I have been able to cancel out a number of terms, but have some left that I can't simplify.

In the expression below there are no subscripts, everything is superscript:

9^(n-1).(-2)^(n+1) + (-2)^(n-1).9^(n+1) - 2(9^n (-2)^n)

is it possible to simplify this any further?


0
Comment
Question by:purplesoup
  • 4
  • 3
8 Comments
 
LVL 32

Expert Comment

by:phoffric
ID: 35124693
>> LHS has Un+1     Un-1 - (Un)^2
                           ^
                           |
             are you missing an operator here?


>> 9^(n-1).(-2)^(n+1) + (-2)^(n-1).9^(n+1) - 2(9^n (-2)^n)
>> is it possible to simplify this any further?
   You can do some simplifying

Rules:
a^(b+c) = a^b a^c
a^(b-c) = a^b a^(-c) =  a^b / a^c

For example, 9^(n+1) = 9^n * 9    and    (-2)^(n-1) = (-2)^n / (-2)^(1)

You should end up with some common factors.
0
 
LVL 37

Expert Comment

by:TommySzalapski
ID: 35125459
I work out the equation and end up with a 121 (which is 11^2) and a bunch of -18s. That doesn't look like it'll work out well. If you get the same thing and it isn't right, then maybe something is amiss earlier in the equation.
0
 

Author Comment

by:purplesoup
ID: 35128709
Sorry I still can't finish it. Applying the above rules I get this.

a^(b+c) = a^b a^c
a^(b-c) = a^b a^(-c) =  a^b / a^c

I get the following:

9^(n-1).(-2)^(n+1) + (-2)^(n-1).9^(n+1) - 2(9^n (-2)^n)

= (9^n.(-2)^n.(-2))/9 + ((-2)^n.9^n.9)/(-2) - 2.9^n.(-2)^n

= 9^n.(-2)^n (-2/9 - 9/2 -2)

= 9^n.(-2)^n (59/18)

???

Where did I go wrong?
0
Independent Software Vendors: We Want Your Opinion

We value your feedback.

Take our survey and automatically be enter to win anyone of the following:
Yeti Cooler, Amazon eGift Card, and Movie eGift Card!

 
LVL 32

Expert Comment

by:phoffric
ID: 35129008
= 9^n.(-2)^n (-2/9 - 9/2 -2)
= 9^n.(-2)^n (-121/18)


0
 

Author Comment

by:purplesoup
ID: 35129102
doh - sorry.

Is 9^2 . (-2)^2 = (-18)^n ??

0
 
LVL 32

Expert Comment

by:phoffric
ID: 35129135
>> Is 9^2 . (-2)^2 = (-18)^n ??
LHS has no n
RHS has n
So I'm not sure what you mean unless you were trying to solve for n.

>> 9^2 . (-2)^2 = 9*9 * (-2)*(-2) = 81 * 4 = 324
but something tells me that this is not what you are looking for.
0
 

Author Comment

by:purplesoup
ID: 35129155
Sorry I typed it wrong, I was trying to simplify, I meant

Is 9^n . (-2)^n = (-18)^n ??

so the final expression would be

= (-18)^n (-121/18)

= -121. (-18)^(n-1)
0
 
LVL 32

Accepted Solution

by:
phoffric earned 2000 total points
ID: 35129576
(ab)^n = a^n  b*n

so,  (-18)^n  =  (-2 * 9)^n  = (-2)^n 9^n

===

9^n.(-2)^n (-121/18) = (-18)^n (-121/18) = -121* (-18)^(n-1)


0

Featured Post

[Webinar] Cloud Security

In this webinar you will learn:

-Why existing firewall and DMZ architectures are not suited for securing cloud applications
-How to make your enterprise “Cloud Ready”, and fix your aging DMZ architecture
-How to transform your enterprise and become a Cloud Enabler

Question has a verified solution.

If you are experiencing a similar issue, please ask a related question

Foreword (May 2015) This web page has appeared at Google.  It's definitely worth considering! https://www.google.com/about/careers/students/guide-to-technical-development.html How to Know You are Making a Difference at EE In August, 2013, one …
We are taking giant steps in technological advances in the field of wireless telephony. At just 10 years since the advent of smartphones, it is crucial to examine the benefits and disadvantages that have been report to us.
Finds all prime numbers in a range requested and places them in a public primes() array. I've demostrated a template size of 30 (2 * 3 * 5) but larger templates can be built such 210  (2 * 3 * 5 * 7) or 2310  (2 * 3 * 5 * 7 * 11). The larger templa…
I've attached the XLSM Excel spreadsheet I used in the video and also text files containing the macros used below. https://filedb.experts-exchange.com/incoming/2017/03_w12/1151775/Permutations.txt https://filedb.experts-exchange.com/incoming/201…
Suggested Courses

916 members asked questions and received personalized solutions in the past 7 days.

Join the community of 500,000 technology professionals and ask your questions.

Join & Ask a Question