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

This topic has 3 expert replies and 0 member replies

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

Post Thu Sep 14, 2017 2:10 pm
If 3^x - 3^(x-1) = 162, then x(x - 1) =

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

The OA is C.

Can some expert do the calculus for me? Thanks.

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Top Reply
Post Tue Sep 19, 2017 2:48 pm
Vincen wrote:
If 3^x - 3^(x-1) = 162, then x(x - 1) =

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

The OA is C.
We can simplify the left side of the equation by factoring out a common 3^x from both terms, and then factor 162 as 3^4 * 2^1. Then we have:

3^x - 3^x * 3^-1 = 3^4 * 2^1

3^x(1 - 3^-1) = 3^4 * 2^1

On the left side, note that the expression 1 - 3^-1 = 1 - 1/3 = 2/3. We now have:

3^x(2/3) = 3^4 * 2^1

3^x = (3^4 * 2)(3/2)

3^x = 3^4 *3

3^x = 3^5

x = 5

So, x(x-1) = 5(4) = 20.

Alternate Solution:

Note that 3^x = 3 * 3^(x - 1). Then the left hand side of the equation becomes:

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

Let’s factor the common 3^(x - 1):

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

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

3^(x - 1)(2) = 162

3^(x - 1) = 81

3^(x - 1) = 3^4

x - 1 = 4

x = 5

Then, x(x - 1) = 20.

Answer: C

_________________
Jeffrey Miller Head of GMAT Instruction

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Top Reply
Post Thu Sep 14, 2017 5:52 pm
Vincen wrote:
If 3^x - 3^(x-1) = 162, then x(x - 1) =

A. 12
B. 16
C. 20
D. 30
E. 81
3^x and 3^(x-1) are TWO CONSECUTIVE POWERS OF 3.
Since 3^x - 3^(x-1) = 162, we need two consecutive powers of 3 that have a difference of 162.

List powers of 3:
3¹ = 3
3² = 9
3³ = 27
3⁴ = 81
3⁵ = 243
.

The two options in blue have a difference of 162:
243-81 = 162.
Thus:
3^x = 3⁵ = 243
3^(x-1) = 3⁴ = 81.

Since x=5 and x-1=4, we get:
x(x-1) =5*4 = 20.

The correct answer is C.

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GMATGuruNY@gmail.com
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For more information, please email me at GMATGuruNY@gmail.com.

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Post Tue Sep 19, 2017 2:48 pm
Vincen wrote:
If 3^x - 3^(x-1) = 162, then x(x - 1) =

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

The OA is C.
We can simplify the left side of the equation by factoring out a common 3^x from both terms, and then factor 162 as 3^4 * 2^1. Then we have:

3^x - 3^x * 3^-1 = 3^4 * 2^1

3^x(1 - 3^-1) = 3^4 * 2^1

On the left side, note that the expression 1 - 3^-1 = 1 - 1/3 = 2/3. We now have:

3^x(2/3) = 3^4 * 2^1

3^x = (3^4 * 2)(3/2)

3^x = 3^4 *3

3^x = 3^5

x = 5

So, x(x-1) = 5(4) = 20.

Alternate Solution:

Note that 3^x = 3 * 3^(x - 1). Then the left hand side of the equation becomes:

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

Let’s factor the common 3^(x - 1):

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

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

3^(x - 1)(2) = 162

3^(x - 1) = 81

3^(x - 1) = 3^4

x - 1 = 4

x = 5

Then, x(x - 1) = 20.

Answer: C

_________________
Jeffrey Miller Head of GMAT Instruction

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Thanked by: Vincen

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