If \(5^x - 5^{x-1}= 500\), what is the value of (x - 1)^2?

(A) 1
(B) 4
(C) 9
(D) 25
(E) 36


Kudos for a correct solution.
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5^x - 5^(x-1) = 500
=> 5^(x-1) {5 - 1} =500
=> 5^(x-1) 4 = 500
=> 5^(x-1) 2^2 = 5^3 2^2
=> 5^(x-1) = 5^3
=> x-1 = 3
=> (x-1)^2 = 3^2 = 9

Answer: C

\(5^x - 5^{x-1}= 500\)

or \(5^x - \frac{5^x}{5}= 500\)

or \(5^x(1 - \frac{1}{5})= 500\)

or \(5^x( \frac{4}{5})= 125*4\)

or \(5^{x-1}\) = 125

or \(5^{x-1}= 5^3\)

or \(x-1 = 3\)

or \((x-1)^2 = 9\)

Answer:- C

Testing the answer choices could also be pretty quick.

A) 1
This means (x - 1)^2 = 1
So, x - 1 = 1, which mean x = 2
Now plug x = 2 into \(5^x - 5^{x-1}= 500\) to get \(5^2 - 5^{2-1}= 500\)
Simplify: 25 - 5 = 500
Doesn't work


B) 4
This means (x - 1)^2 = 4
So, x - 1 = 2, which mean x = 3
Now plug x = 3 into \(5^x - 5^{x-1}= 500\) to get \(5^3 - 5^{3-1}= 500\)
Simplify: 125 - 25 = 500
Doesn't work

C) 9
This means (x - 1)^2 = 9
So, x - 1 = 3, which mean x = 4
Now plug x = 4 into \(5^x - 5^{x-1}= 500\) to get \(5^4 - 5^{4-1}= 500\)
Simplify: 625 - 125 = 500
WORKS!

Answer: C

Cheers,
Brent
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Hi All,

This question has a great 'brute force' shortcut to it (if you're comfortable doing some basic multiplication).

Since the answer choices are all perfect squares, X MUST be an integer.

We're also given 5^X and 5^(X-1), which are two consecutive "powers" of 5. We're told that subtracting the smaller value from the larger value will give us 500...

Let's start listing powers of 5 until we find two consecutive powers that differ by 500....

5^0 = 1
5^1 = 5
5^2 = 25
5^3 = 125
5^4 = 625

STOP. 625 - 125 = 500, so X MUST be 4. To confirm...

5^4 - 5^(4-1) =
5^4 - 5^3 =
625 - 125 = 500

Since we now know the value of X, we can answer the question - the value of (X-1)^2 = (4-1)^2 = 9

Final Answer:

GMAT assassins aren't born, they're made,
Rich
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To be honest I have tried to memorize the lowest numbers and their exponents up to around 6-8 (depending on the number i.e. 2^8 but not 5^8) as that was a tip from the Manhattan Guides.

As soon as I saw 5^x I just thought: 5, 25, 125, 625 and instantly noticed that 625-125 = 500.

But I can see the merit in the other way of factoring as the questions may not all have an easy solution like this one!

(5^x) - (5^x)*5 = 500
5^x (1-5) = 500
5^x = -125
5^x = -5^3
x=3 => (3-1)^2 = 4