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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.

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The answer I get is not one of your options. Have I misinterpreted the question? –  Austin Mohr Nov 30 '12 at 7:22
    
I just looked up and browsed through an NMAT practice exam, and every question on it had 5 options. Are you sure that this one only has 4? –  rayradjr Nov 30 '12 at 7:49
    
thanks for replying. I think Henry figured it out. I think the + sign is supposed to be a divide symbol. Solving it that way gives an answer of 3/8. Sorry, for the trouble. –  A.G. Nov 30 '12 at 7:52
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1 Answer

up vote 5 down vote accepted

If you are allowed to assume a typo of $+$ for the similar $\div$, you can get one of the choices:

$$3^{n+2}\div(3^{n+3}-3^{n+1}) = \frac38.$$

Since all the choices have denominators, you need to do a division or use negative exponents to get them.

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That was my notion, too; unfortunately, it’s the wrong choice, and I can’t see any reasonable typo or OCR error that would make $\frac13$ the right choice. I agree that the answer choices pretty much guarantee that the original problem had a division or negative exponent somewhere. –  Brian M. Scott Nov 30 '12 at 7:42
    
Thank you sir! I think you are correct. I feel embarrassed for having asked this question. Thanks again. –  A.G. Nov 30 '12 at 7:42
1  
@Austin Mohr: But $3^{n+2}\div(3^{n+3}-3^{n+1})=\dfrac{3^{n+1}\times 3}{3^{n+1}\times (3^2-1)}$ –  Henry Nov 30 '12 at 8:00
    
@BrianM.Scott Thanks for the reply. The answers for the practice exam are listed on a separate page. Only the letter for the correct answer is given. I think D was typed instead C by accident. sorry for the trouble. –  A.G. Nov 30 '12 at 8:18
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