Discrete Math: Recurrence Relations Explanation

Question

I was looking over one of the solutions for one of my homework assignments and was a little confused as to how they factored part of the solution.

Solution

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I'm a little confused as to how they turned this

$8((n-1)4^{n-1}))-16((n-2)4^{n-2}))$

into

$(n-1)2^3(2^{2n-2})-(n-2)2^4(2^{2n-4})$

Could anyone explain this to me a little better?

Answer 1

See that

$8=2\cdot2\cdot2=2^3$

$16=2\cdot2\cdot2\cdot2=2^4$

$4^{n-1}=(2\cdot2)^{n-1}=(2^2)^{n-1}=2^{2(n-1)}=2^{2n-2}$.

$4^{n-2}=(2\cdot2)^{n-2}=(2^2)^{n-2}=2^{2(n-2)}=2^{2n-4}$.

using these identities you get the desired result.