Solved

Simplifying Linear Second Order Recurrence sequences...

Posted on 2011-03-13
8
507 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
[X]
Welcome to Experts Exchange

Add your voice to the tech community where 5M+ people just like you are talking about what matters.

  • Help others & share knowledge
  • Earn cash & points
  • Learn & ask questions
  • 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
Industry Leaders: 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 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

Industry Leaders: 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!

Question has a verified solution.

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

Suggested Solutions

Title # Comments Views Activity
Group Data Frequency Distribution 9 53
Revenue table 8 135
Graph 2 121
x2 – x2 = x2 – x2 conundrum 12 79
This article provides a brief introduction to tissue engineering, the process by which organs can be grown artificially. It covers the problems with organ transplants, the tissue engineering process, and the current successes and problems of the tec…
When we purchase storage, we typically are advertised storage of 500GB, 1TB, 2TB and so on. However, when you actually install it into your computer, your 500GB HDD will actually show up as 465GB. Why? It has to do with the way people and computers…
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.
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…

751 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