[email protected] wrote:Is 5^k less than 1,000?
(1)5^(k+1) > 3,000
(2)5^(k-1) = 5^k - 500

Here's another approach:

NOTE: I should mention that shibsriz's original wording is misleading. The question has been edited to correct any ambiguitiy.

Target question: Is 5^k less than 1000?

Statement 1: 5^(k+1) > 3000
First notice that 5^(k+1) = (5^k)(5^1)
So, we can take 5^(k+1) > 3000 and divide both sides by 5 to get: 5^k > 600
There are several possible cases to consider. Here are two:
case a: 5^k = 601, in which case 5^k is less than 1000.
case b: 5^k = 1001, in which case 5^k is not less than 1000.
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT.

Statement 2: 5^(k-1) = 5^k - 500
IMPORTANT: Notice that we're given an EQUATION, which means we can solve the equation to find the definitive value of k. If we can find the value of k, then we can instantly tell whether or not 5^k is less than 1000. So, it SEEMS that we can conclude that statement 2 is sufficient WITHOUT performing any calculations. HOWEVER, if it's the case that the equation yields 2 possible values of k, then it may be the case that one value of k is such that 5^k is less than 1000, and the other value of k is such that 5^k is greater than 1000. So, at this point, we need only determine whether or not the equation will yield 1 or 2 values of k.

Rearrange to get the k's on one side: (5^k) - 5^(k-1) = 500
Factor the left side: 5^(k-1)[5 - 1] = 500
Simplify: 5^(k-1)[4] = 500
STOP!!
At this point, we can see that this equation will yield only one value of k. So, IF WE WERE to solve the equation for k, we would definitely be able to determine whether or not 5^k is less than 1000.
Since we can answer the target question with certainty, statement 2 is SUFFICIENT.

Answer = B

Cheers,
Brent

freyesinsb wrote:

Rearrange to get the k's on one side: (5^k) - 5^(k-1) = 500
Factor the left side: 5^(k-1)[5 - 1] = 500
Simplify: 5^(k-1)[4] = 500

Brent, How did you get to this point?

The factoring that I performed above is the same as these examples:
x^9 - x^8 = x^8(x - 1)
j^3 - j^2 = j^2(j - 1)
q^15 - q^14 = q^14(q - 1)
Notice that each expression features two terms, and each time I factored out the term with the smaller exponent.

Likewise, in the expression (5^k) - 5^(k-1), the exponent (k-1) is smaller, so we can factor our 5^(k-1) to get:
(5^k) - 5^(k-1) = 5^(k-1)[5 - 1]
= 5^(k-1)[4]

Cheers,
Brent