# Having a problem simplifying the following

• Dec 31st 2012, 06:05 PM
exp13
Having a problem simplifying the following
Has left me so very confused:

Un+1= 5^n+1+(-8)^n+1
Un-1= 5^n-1+(-8)^n-1
Un^2 =5^2n+2(5^n(-8)^n)+(-8)^2n
Un+1Un-1 = 5^2n+5^n+1(-8)^n-1+5^n-1(-8)^n+1+(-8)^2n
after cancelling
Un+1Un-1-Un^2 = 5^2n + 5^n+1(-8)^n-1 + + 5^n-1(-8)^n+1 +(-8)^2n - (5^2n + 2(5^n(-8)^n) + (-8)^2n
= 5^n+1(-8)^n-1 + 5^n-1(-8)^n+1 - 2(5^n(-8)^n)
= 5^n-1(-8)^n-1 * (5^2+(-8)^2 - (2*5*(-8))
= (-40)^n-1 * (89-(-80))
= 169(-40)^n-1

If someone could please explain how the simplifications/cancellations (shown in red, previous steps are fine) have occurred, it would be much appreciated!
• Dec 31st 2012, 06:40 PM
Wilmer
Re: Having a problem simplifying the following
Quote:

Originally Posted by exp13
Has left me so very confused:

Un+1= 5^n+1+(-8)^n+1
Un-1= 5^n-1+(-8)^n-1
Un^2 =5^2n+2(5^n(-8)^n)+(-8)^2n
Un+1Un-1 = 5^2n+5^n+1(-8)^n-1+5^n-1(-8)^n+1+(-8)^2n
after cancelling
Un+1Un-1-Un^2 = 5^2n + 5^n+1(-8)^n-1 + + 5^n-1(-8)^n+1 +(-8)^2n - (5^2n + 2(5^n(-8)^n) + (-8)^2n
= 5^n+1(-8)^n-1 + 5^n-1(-8)^n+1 - 2(5^n(-8)^n)
= 5^n-1(-8)^n-1 * (5^2+(-8)^2 - (2*5*(-8))
= (-40)^n-1 * (89-(-80))
= 169(-40)^n-1

If someone could please explain how the simplifications/cancellations (shown in red, previous steps are fine) have occurred, it would be much appreciated!

The steps in black are fine? Are you joking? Still celebrating New Year? (Happy)

Your expression is quite unclear; like 5^n+1 means what?
5 to the power (n+1) or (5^n) + 1

If the steps in black are fine, then how did you get from:
Un-1= 5^n-1+(-8)^n-1
to:
Un^2 =5^2n+2(5^n(-8)^n)+(-8)^2n ??????
• Dec 31st 2012, 06:43 PM
exp13
Re: Having a problem simplifying the following
Sorry, I should really explain more about the context of the question which is:

Let un be the sequence, un=5n+(-8)n?
where n=0,1,2,...... ?

Show that un satisfies the identity
un+1un-1-un^2=169(-40)^n-1, For n=1,2,3,.....

Hence the black part seems to be in line with other examples I've seen
• Dec 31st 2012, 11:50 PM
ibdutt
Re: Having a problem simplifying the following
• Jan 1st 2013, 07:04 AM
skeeter
Re: Having a problem simplifying the following