The problem is from an NMAT Practice Exam. The problem is multiple choice. It looks easy enough...
$3^{n+2}+(3^{n+3}-3^{n+1}) =~?$
a.) $\dfrac1{3^{n+1}}$
b.) $\dfrac1{3^{n+2}}$
c.) $\dfrac38$
d.) $\dfrac13$
The answer given is $\frac13$, but I don't know how they got that.
My attempts:
$3^{n+2}+(3^{n+3}-3^{n+1})=3^n(9+27-3)=33\cdot3^n$
Another attempt using self similarity...
$y=3^{n+2}+(3^{n+3}-3^{n+1})$
$3y=3^{n+3}+(3^{n+4}-3^{n+2})$
$3y-y=3^{n+4}-2\cdot3^{n+2}+3^{n+1}$
I'm trying help someone out with the math section, but I'm lost on how to solve this one. Thanks in advance.