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!

Quote:

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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!

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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 ??????

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

Attachment 26432

duplicate post ...

http://mathhelpforum.com/advanced-al...following.html