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

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

Can you not just work this out?

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

Therefore x is 4 so (x-1)^2 = (4-1)^2 = 9 (c)

This way just seems quicker to me!

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

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

How did you "work it out"?
If you guess and tested, that's totally fine, except that approach could take a while.

Cheers,
Brent

I believe the answer is C. Please see below for explanation

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

5^x(1 - 1/5 ) = 5* 2*5 * 2*5

5^x(4/5) = 5^3 * 2^2

5^x * 4 = 5^4 * 2^2

5^x * 2^2 = 5^4 * 2^2

Which means that x = 4

Replacing x with 4 in (x - 1)^2 gives 9


Answer C

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

\(5^x−5^{x-1}\)=500

This can be simplified to

\(5^x\) * (4/5) =\(5^3\) * \(2^2\)


\(5^{x-1}\) * \(2^2\) = \(5^3\) * \(2^2\)

x - 1 = 3


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

Answer = C

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!

Simplify both sides to their basic roots

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

L.H.S R.H.S
\(5^{x-1}(5-1)\) = \(5^3 * 2^2\)
\(5^{x-1}(4)\) = \(5^3 * 2^2\)
\(5^{x-1}(4)\) = \(5^3 * 2^2\)

Therefore
x-1 = 3
Square both sides
\((x-1)^2\) = 9

Cheers

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

Notice that 5^x = 5 * 5^(x - 1). Thus, we can write:

\(\Rightarrow\) 5^x - 5^(x - 1) = 500

\(\Rightarrow\) 5 * 5^(x - 1) - 5^(x - 1) = 500

\(\Rightarrow\) 5^(x - 1)[5 - 1] = 500

\(\Rightarrow\) 5^(x - 1) * 4 = 500

\(\Rightarrow\) 5^(x - 1) = 125

\(\Rightarrow\) 5^(x - 1) = 5^3

Equating the exponents of the expressions on each side of the equation, we get x - 1 = 3, which means x = 4. Thus, (x - 1)^2 = 3^2 = 9.

Answer: C

Given that \(5^x - 5^{x-1}= 500\) and we need to find the value of (x - 1)^2

\(5^x - 5^{x-1}\) = 500
=> \(5^{x-1 + 1} - 5^{x-1}\) = 500
=> \(5^{x-1} * 5^1 - 5^{x-1}\) = 500
=> \(5^{x-1} * (5 - 1)\) = 500
=> \(5^{x-1} = \frac{500}{4}\) = 125 = \(5^3\)
=> x - 1 = 3
=> x = 4

=> \((x - 1)^2\) = \((4 - 1)^2\) = 9

So, Answer will be C
Hope it helps!

Watch the following video to learn the Basics of Exponents