# Any ideas around simplifying the following?

• Dec 31st 2012, 06:06 PM
exp13
Any ideas around 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!
• Jan 1st 2013, 12:44 AM
earboth
Re: Any ideas around 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!

1. A happy new year to you!

2. Your calculations are really hard to understand because you are so mean in using brackets ...

3. The blue and green marked terms add up to zero. That's all.

4. to **: Factor out $\displaystyle 5^{n-1} \cdot (-8)^{n-1}$ of each of the three terms

5. to ***: Evaluate the sum. Use the property: $\displaystyle a^x \cdot b^x = (a \cdot b)^x$