<s>[INST] Show that $(n - 1)2^{n+1} + 2 + (n+1)2^{n+1} = n(2^{n+2})+2$ I'm having a really hard time showing this equality is true, I've tried several ways of going about it and I just can't seem to make it work. Help!
$(n - 1)2^{n+1} + 2 + (n+1)2^{n+1} = n(2^{n+2})+2$
Thanks!
 [/INST] Just let $A=2^n$. Then $2^{n+1}=2\cdot 2^n=2A$ and $2^{n+2}=2^2\cdot 2^n = 4A$. Then your equality reads:
$$2(n - 1)A + 2 + 2(n+1)A = 4nA+2.$$
Let us show that this is indeed true:
$$2(n - 1)A + 2 + 2(n+1)A = 2A(n-1+n+1)+2=2A(2n)+2=4nA+2,$$
as desired.
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