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How do I simplify this expression?

https://imgur.com/a/mISQZN2
So first off, I have this solution for an. But, I want to simplify it for an+1 as well. I use 6(2^n + 2^2n) - 8(2^(n-1) + 2^(2n-2)) but I just do not get the right answer, I end up getting 2^(2n + 2) + 3*2^(n+1) - 2^(n+2) and I have no idea why. It should be the same just the index is different, but I just cannot get rid of the 3
(edited 4 months ago)
From where you left off,
32n+12n+2=32n+122n+13 \cdot 2^{n+1} - 2^{n+2} = 3 \cdot 2^{n+1} - 2 \cdot 2^{n+1}.
So it's the same as what you want.
(Moral of the story, look at what your perceived goal is, and "make it happen" - here our goal don't have 2n+22^{n+2})

Though I don't see why you have to find an+1a_{n+1} all over again from scratch, when you've already done the hard work to simplify ana_{n}.
(edited 4 months ago)

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