If 4a + 4a+1 = 4a-2 -176, what is the value of a?

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If 4a + 4a+1 = 4a-2 -176, what is the value of a?

by pythuy » Tue Nov 19, 2019 3:31 am
If $$4^a$$ + $$4^{a+1}$$ = $$4^{a+2}$$ -176, what is the value of a?

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by [email protected] » Fri Nov 22, 2019 9:47 am
Hi pythuy,

To start, when posting specific GMAT questions, you should make sure to post them in the proper sub-forum. For example, the Problem Solving Forum can be found here:

https://www.beatthegmat.com/problem-solving-f6.html

In addition, you should post the FULL prompt (including the 5 answers choices and the correct answer).

With this question, we're told that 4^A + 4^(A+1) = 4^(A+2) - 176. We're asked for the value of A. If we had the 5 answer choices, then we could TEST THE ANSWERS and find the number that properly "fits" the given equation. Even without that information though, solving this problem isn't too tough - we can use 'brute force' to do it.

Here are the first several 'powers of 4':
4^1 = 4
4^2 = 16
4^3 = 64
4^4 = 256
4^5 = 1024

Notice that as the exponent increases, the difference in the values between consecutive powers of 4 significantly increases. That's an important detail, since this question asks us to deal with 3 consecutive powers of 4 AND a difference of 176. Looking at that list of numbers, we can see a difference of 192 between 4^3 and 4^4, so that's a clue as to where we should be looking.

IF..... A = 2....
4^2 = 16, 4^3 = 64 and 4^4 = 256

Does 16 + 64 = 256 - 176?
80 = 80
This equation is correct, so A=2 must be the answer.

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Rich
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by nitink » Wed Nov 27, 2019 10:38 pm
4^A + 4^(A+1) = 4^(A+2) - 176

Take 4^A common from LHS, 4^A (1+ 4) = 4^A . 4^2 - 11. 4^2

take 4^2 common from RHS, 5. 4^A = 16 (4^A -11)

(5/16). 4^ A = 4^A -11

11= 4^A (1 - 5/16)

11= 4^A . 11/16

16= 4^A

4^A = 4^2

so, A=2

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