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Math Help - Any ideas around simplifying the following?

  1. #1
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    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!
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  2. #2
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    Re: Any ideas around simplifying the following?

    Quote Originally Posted by exp13 View Post
    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 5^{n-1} \cdot (-8)^{n-1} of each of the three terms

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