Thread: 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!

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 ??????

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

duplicate post ...

Any ideas around simplifying the following?