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(1) 5^(k+1) > 3,000 --> \(5^k>600\) --> if \(k=4\) then the answer is YES: since \(600<(5^4=625)<1,000\) but if \(k=10\), for example, then the answer is NO. Not sufficient.

(2) 5^(k-1) = 5^k - 500 --> we can solve for k and get the single numerical value of it, hence this statement is sufficient. Just to illustrate: \(5^k-5^{k-1}=500\) --> factor out \(5^{k-1}\): \(5^{k-1}(5-1)=500\) --> \(5^{k-1}=125\) --> \(k-1=3\) --> \(k=4\). Sufficient.

Statement(1) : 5^(k+1) > 3000 The above inequality can be reduced to 5^k > 600. From this, we clearly know few possible values for k i.e., 4,5,6,.. Substituting these values in the inequality given in the question gives away both yes and no answers. k = 4, 5^(4-1) < 1000 k = 5, 5^(5-1) < 1000 k = 6, 5^(6-1) > 1000 Hence statement(1) is not sufficient.

Statement(2): 5^(k-1) = 5^k - 500 Reducing the above inequality, 4/5 * 5^k = 500 So 5^k = 625 = 5^4. Clearly k = 4 and the original inequality is satisfied: 5^4 < 1000. Hence statement(2) is sufficient.

The biggest take-away here should be that we don't need to solve the St-2. The moment you start solving St2, you have fallen for GMAT's classic time waster trap. See below.

Statement1: As 5^(k+1) > 3,000 --> k>4 and hence insufficient Statement2: We dont need to solve the equation. Since this is an EQUATION (and not an inequality) with one variable, we will get the exact value of k and we will be able to answer the question one way or the other. SUFFICIENT.
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My journey V46 and 750 -> http://gmatclub.com/forum/my-journey-to-46-on-verbal-750overall-171722.html#p1367876

(1) 5^(k+1) > 3,000 --> \(5^k>600\) --> if \(k=4\) then the answer is YES: since \(600<(5^4=625)<1,000\) but if \(k=10\), for example, then the answer is NO. Not sufficient.

(2) 5^(k-1) = 5^k - 500 --> we can solve for k and get the single numerical value of it, hence this statement is sufficient. Just to illustrate: \(5^k-5^{k-1}=500\) --> factor out \(5^{k-1}\): \(5^{k-1}(5-1)=500\) --> \(5^{k-1}=125\) --> \(k-1=3\) --> \(k=4\). Sufficient.

Answer: B.

Hope it's clear.

Hi can we generalize that " Every time when there is an equation with exponential expressions and there is only one single variable exponent(and no other variable in equation), we can always find the value of that exponent." Is there any exception possible?

Right from the beginning, there are 2 things that you should note about this DS prompt:

1) At NO POINT does it state that K has to be an integer. 2) It's clearly based on exponents, so some exponent rules/patterns MUST be involved.

We're asked if 5^K is < 1,000. This is a YES/NO question.

Fact 1: 5^(K+1) > 3,000

In this Fact, notice how the exponent (K+1) differs from the exponent in the question (K). There's an exponent rule that accounts for this difference.

As an example, consider... 5^2 = 25 5^3 = 125 Notice how 5^3 is "5 times" greater than 5^2? This difference occurs because the base is 5 and we're increasing the exponent by 1. It can also be used in reverse....

5^3/5^2 = 5^(3-2) = 5^1 = 5

This is a standard rule about "dividing" exponents with the same base --> we SUBTRACT the exponents.

With Fact 1, we're dealing with 5^(K+1) and the question is dealing with 5^K. This means that DIVIDING 5^(K+1) by 5 will give us 5^K:

5^(K+1)/5^1 = 5^(K+1-1) = 5^K.

This is all meant to say that we can DIVIDE both sides of this inequality by 5, which gives us...

5^(K+1) > 3,000 5^K > 600

IF.... 5^K = 601 then the answer to the question is YES 5^K = 1,001 then the answer to the question is NO Fact 1 is INSUFFICIENT

Fact 2: 5^(K-1) = 5^K - 500

This is a 1 variable, 1 equation "system", so we CAN solve it (and there will only be 1 answer). Even if you did not know that, it's still easy enough to get to the solution.... Since most Test Takers are better at basic multiplication than they are at manipulating higher-level exponents, here's how you can "brute force" the solution:

5^1 = 5 5^2 = 25 5^3 = 125 5^4 = 625 5^5 = 3125

Find two consecutive powers of 5 that differ by 500 and you have the solution to the above equation. 5^4 - 5^3 = 625 - 125 = 500 Fact 2 is SUFFICIENT.

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Here the question will be fairly easy if we consider it as an inequality expression rather than Exponents as never does it state that K is an integer..
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Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
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