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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
Elapsed Time: 00:00
  • Lap #[LAPCOUNT] ([LAPTIME])
    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.

    Need free GMAT or MBA advice from an expert? Register for Beat The GMAT now and post your question in these forums!
    Post Thu Sep 14, 2017 3:11 pm
    Vincen wrote:
    If 3^x - 3^(x-1) = 162, then x(x - 1) =

    A. 12
    B. 16
    C. 20
    D. 30
    E. 81
    Given: 3^x - 3^(x-1) = 162
    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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    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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    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

    Thanked by: Vincen

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