Problem Solving

If (3^x) - 3^(x-1) = 162, then x(x-1) =

A) 12
B) 16
C) 20
D) 30
E) 81

Answer is C, but exactly how?

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

 (3^(x-1))(3 -1) = 162
 (3^(x-1))2 = 162
 (3^(x-1)) = 162/2 = 81 = 3^4
 x-1 = 4
 x = 5
Therefore, x(x-1) = 4*5 = 20…………… your answer

Where did the (3-1) come from?

musicdaemon wrote:Since, (3^x) - 3^(x-1) = 162

the above equation can also be written as:

3x3^(x-1) - 3^(x-1) = 162
taking 3^(x-1) common
we get:

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

musicdaemon wrote:  (3^(x-1))(3 -1) = 162
 (3^(x-1))2 = 162
 (3^(x-1)) = 162/2 = 81 = 3^4
 x-1 = 4
 x = 5
Therefore, x(x-1) = 4*5 = 20…………… your answer

I am very confused

When factoring out the common factor how are you getting 3x3^x-1 - 3^x-1=162

Isnt the original question 3^x-3^x-1=162 So wouldn't the common factor be 3x can someone please explain step by step.

Thank you,

RB

adi wrote:If (3^x) - 3^(x-1) = 162, then x(x-1) =

A) 12
B) 16
C) 20
D) 30
E) 81

Answer is C, but exactly how?

We can plug in the answers, which represent the value of x(x-1).

Answer choice C: x(x-1) = 20.
If x(x-1) = 20, then x = 5, since 5*(5-1) = 5*4 = 20.
Plugging x=5 into 3^x - 3^(x-1) = 162, we get:
3� - 3^(5-1) = 162
3� - 3� = 2 * 3�
3�(3-1) = 2 * 3�
3� * 2 = 2 * 3�. Success!

The correct answer is C.

Alternatively, the numbers are small enough that we could plug in the answers and expand the exponents:

Answer choice C: x(x-1) = 20.
If x(x-1) = 20, then x = 5, since 5*(5-1) = 5*4 = 20.
Plugging x=5 into 3^x - 3^(x-1) = 162, we get:
3� - 3^(5-1) = 162
3� - 3� = 162
243 - 81 = 162. Success!

162 can be written as 2 * 3^4
& 3^(x-1) can be written as (3^x)/3

now given, 3^x - 3^(x-1) = 2 * 3^4
=> 3^x - (3^x)/3 = 2 * 3^4

Take 3^x common ---> 3^x[ 1-(1/3)] = 2* 3^4
3^x [2/3]== 2* 3^4

cancelling 2 on both sides (3^x)/3 = 3^4
Taking denominator 3 on other side we get 3^x = 3^5
=> x=5 (as bases are same)

now x (x-1) is 5 (5-1) = 5*4 = 20 (C)

Hope it helps!.

Thank you, Thank you soooooooooo much, you provided me with my light bulb moment.

adi wrote:If (3^x) - 3^(x-1) = 162, then x(x-1) =

A) 12
B) 16
C) 20
D) 30
E) 81

Answer is C, but exactly how?

As a follow-up to Mitch's post (backsolving is a great way to solve this question), we can use common sense and a bit of arithmetic to quickly eliminate two of the five choices, making backsolving even more viable.

We know that the correct answer is the product of x and (x-1). Since this is the GMAT, it's reasonable to assume that x is going to be an interger (otherwise the problem would be well beyond the scope of the exam).

So, which answers can be written as the product of two consecutive integers?

A) 12=3*4... keep it
B) 3*4 = 12; 4*5=20... no way to write 16 as the product of 2 consecutive integers - eliminate B.
C) 20 = 4*5... keep it
D) 30 = 5*6... keep it
E) 8*9 = 72; 9*10 = 90... no way to write 81 as the product of 2 consecutive integers - eliminate E.

Since we only have 3 possible choices, we know that the worst case scenario is that we have to plug 2 of them in (if neither of those work, then the third choice is correct by default).

3^x - 3^(x-1) = 162
3^x - 3^x/3 = 162
3^x(1-1/3) = 162
3^x(2/3) = 162
3^x = 162:2/3
3^x = 162*3/2
3^x = 81*3
3^x = 243
3^x = 3^5

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

1. 3^(x) - 3^(x-1) = 162
2. Factor out 3^x ----> 3^(x) {1 - 3^(-1)} = 162
3. Further simplify: 3^(x) {1 - (1/3) ] = 162
4. 3^(x) (2/3) = 162
5. 3^(x) = 243 ---> x = 5
6. x (x-1) = 5 (5 -1)----> 20

OA = C