GMATprep problem solving question

[cheeno]

Hello.

I forgot to take a screenshot of it but I'm sure it's one of the first few questions.

3^x + 3^x-1=162 What is x(x-1)?

If I remember it correctly, the answer was 16. There were no variables in the answer choices. (I guessed 20 on this problem)

Can you help me guys? I know you can get the GCF of this by 3^x(1+1/3) but after that what do you do?

Thanks a lot in advance.

cheeno

[Stuart@KaplanGMAT]

Hello.

I forgot to take a screenshot of it but I'm sure it's one of the first few questions.

3^x + 3^x-1=162 What is x(x-1)?

If I remember it correctly, the answer was 16. There were no variables in the answer choices. (I guessed 20 on this problem)

Can you help me guys? I know you can get the GCF of this by 3^x(1+1/3) but after that what do you do?

Thanks a lot in advance.

cheeno

Actually, 20 is the correct answer.

This is a great question to backsolve. We know that x is going to be an integer (otherwise the question would be WAY too hard for the GMAT), so we know that x(x-1) is going to be the product of two consecutive integers.

So, the answer is going to have to be something like:

6 (which is 2*3)
12 (which is 3*4)
20 (which is 4*5)
30 (which is 5*6)

and so on.

Once we eliminate any answer which doesn't fit this pattern, we just plug the remaining ones in, starting with the middle value (assuming we have it down to 3 or fewer choices).

If we try 20, we get x=5 and x-1=4.

So, that gives us:

3^5 - 3^4 = 162

243 - 81 = 162

Bingo!!

If the answer we checked gave us a number greater than 162, then we need a smaller answer; if the answer we checked gave us a number less than 162, then we need a bigger answer.

We could solve algebraically, but it's pretty high level (and time consuming) math:

3^x + 3^x-1=162

3^x + (3^x)(3^-1) = 162

3^x (1 - 1/3) = 162

(2/3)(3^x) = 162

3^x = (3/2) 162

3^x = 243

(using trial and error)

x = 5

so, x(x-1) = 5(4) = 20

As you can see, even if you understand the math, backsolving is MUCH quicker.

[its_me07]

is it 3^x + 3^x-1=162

or

3^x - 3^x-1=162?

because u r using a neg sign.

[smushkas]

There must be something wrong, since 3^5 + 3^4 = 324.

[joshi.komal]

Hi Cheeno,

The correct question should be :

3^x - 3^x-1=162
or
3^x + 3^x-1=224

Well spotted guys

[Stuart@KaplanGMAT]

I might have just missed the sign; if it's supposed to be +, then the answer would be different, but the solution basicaly the same.

[senthil]

Stuart Kovinsky, while solving U have changed the sign in the step where u have taken the "3 inverse". It shud have been 3^x(1+1/3) instead of 3^x(1-1/3).thereby it wud have become 3^x(4/3)=162,which wud lead to a fraction.
I wud have approached this question like,
3^x-3^(x-1)=162
162=(3^4)2=(3^4)(3-1)=(3^5)-(3^4)
Therefore, 3^x -3^(x-1) =(3^5)-(3^4)
hence comparing x=5.
x(x-1)=20

Thanks
Senthil

[Stuart@KaplanGMAT]

Stuart Kovinsky, while solving U have changed the sign in the step where u have taken the "3 inverse". It shud have been 3^x(1+1/3) instead of 3^x(1-1/3).thereby it wud have become 3^x(4/3)=162,which wud lead to a fraction.
I wud have approached this question like,
3^x-3^(x-1)=162
162=(3^4)2=(3^4)(3-1)=(3^5)-(3^4)
Therefore, 3^x -3^(x-1) =(3^5)-(3^4)
hence comparing x=5.
x(x-1)=20

Thanks
Senthil

Actually, the problem is that the question was supposed to be:

3^x - 3^(x-1) = 162

instead of 3^x + 3^(x-1) = 162

otherwise none of our answers make any sense!

I just intuitively changed the question as I was doing the math, sorry for the confusion.