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.
If 3^x - 3^(x-1) = 162, then x(x - 1) =
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- Brent@GMATPrepNow
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Given: 3^x - 3^(x-1) = 162Vincen wrote:If 3^x - 3^(x-1) = 162, then x(x - 1) =
A. 12
B. 16
C. 20
D. 30
E. 81
Factor to get: [3^(x-1)][3^1 - 1] = 162
Simplify to get: [3^(x-1)][2] = 162
Divide both sides by 2 to get: 3^(x-1) = 81
Rewrite the right side as 3^(x-1) = 3^4
So, x - 1 = 4
This means x = 5
We get x(x - 1) = (5)(5 - 1) = (5)(4) = 20
Answer: C
Cheers,
Brent
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3^x and 3^(x-1) are TWO CONSECUTIVE POWERS OF 3.Vincen wrote:If 3^x - 3^(x-1) = 162, then x(x - 1) =
A. 12
B. 16
C. 20
D. 30
E. 81
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
3� = 81
3� = 243.
The two options in blue have a difference of 162:
243-81 = 162.
Thus:
3^x = 3� = 243
3^(x-1) = 3� = 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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- Jeff@TargetTestPrep
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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: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.
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
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