# Thread: Having a problem simplifying the following

1. ## 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!

2. ## Re: Having a problem simplifying the following

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?

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 ??????

3. ## 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