Problem Solving

djsir007 wrote:3^x-3^x-1=162, then x(x-1)=

12
16
20
30
81

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 - â…“ = â…”. 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

Hi All,

We're told that 3^(X) - 3^(X-1) = 162. We're asked for the value of then (X)(X-1). This question can be solved rather easily with a bit of 'brute force' arithmetic.

Since we're dealing with 'powers of 3', let's map out the first several values:
3^1 = 3
3^2 = 9
3^3 = 27
3^4 = 81
3^5 = 243

We're subtracting two consecutive powers of 3 and ending up with 162. Looking at the list so far, we have an obvious 'pair' of values that fits what we're looking for:

3^5 and 3^4
243 - 81 = 162

Thus, X = 5 and the answer to the question is (5)(5-1) = 20

Final Answer: C

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