Target question: Is 5^k less than 1000?Is 5^k less than 1,000?
(1) 5^(k+1) > 3,000
(2) 5^(k-1) = 5^k - 500
Statement 1: 5^(k+1) > 3000
Notice that 5^1 = 5, 5^2 = 25, 5^3 = 125, 5^4 = 625, 5^5 = 3125...
There are several values of k that satisfy the condition that 5^(k+1) > 3000. Here are two:
case a: k = 4 [so, 5^(k+1) = 5^5 = 3125, which is greater than 3000]. In this case 5^k = 5^4 = 625, and 625 IS less than 1000.
case b: k = 5 [so, 5^(k+1) = 5^6 = 15625, which is greater than 3000]. In this case 5^k = 5^5 = 3125, and 3125 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
