Solved

Simplifying Linear Second Order Recurrence sequences...

Posted on 2011-03-13
8
505 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
Announcing the Most Valuable Experts of 2016

MVEs are more concerned with the satisfaction of those they help than with the considerable points they can earn. They are the types of people you feel privileged to call colleagues. Join us in honoring this amazing group of Experts.

 
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 500 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

Live: Real-Time Solutions, Start Here

Receive instant 1:1 support from technology experts, using our real-time conversation and whiteboard interface. Your first 5 minutes are always free.

Question has a verified solution.

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

Suggested Solutions

Title # Comments Views Activity
Calculator Question 8 47
Representing TIME in Excel 8 51
Octane 98 vs 95 petrol / gasoline : mileage,  pros & cons 10 123
Restarting the Universe - A Thought Experiment 19 86
Introduction On a scale of 1 to 10, how would you rate our Product? Many of us have answered that question time and time again. But only a few of us have had the pleasure of receiving a stack of the filled out surveys and being asked to do somethi…
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 …
This is a video describing the growing solar energy use in Utah. This is a topic that greatly interests me and so I decided to produce a video about it.
Although Jacob Bernoulli (1654-1705) has been credited as the creator of "Binomial Distribution Table", Gottfried Leibniz (1646-1716) did his dissertation on the subject in 1666; Leibniz you may recall is the co-inventor of "Calculus" and beat Isaac…

816 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

Need Help in Real-Time?

Connect with top rated Experts

11 Experts available now in Live!

Get 1:1 Help Now