[GMAT math practice question]
$$If\ \ 2^n+2^{n-2}=5120,\ then\ n=?$$
A. 8
B. 9
C. 10
D. 11
E. 12
If 2^n+2^n-2=5120, then n=?
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- Max@Math Revolution
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We can PLUG IN THE ANSWERS, which represent the value of n.Max@Math Revolution wrote:[GMAT math practice question]
$$If\ \ 2^n+2^{n-2}=5120,\ then\ n=?$$
A. 8
B. 9
C. 10
D. 11
E. 12
When the correct answer is plugged in, the given expression will be equal to 5120.
D: n=11
Plugging n=11 into the given expression, we get:
2¹¹ + 2¹¹¯² = 2¹¹ + 2� = 2�(2² + 1) = 512(5) = 2560.
Since the result is too small, the value of n must be GREATER than 11.
The correct answer is D.
To solve algebraically, factor out the SMALLER EXPONENT.
Here, the smaller exponent is n-2:
$$\\2^n+2^{n-2}=5120\\$$
$$\\2^{n-2}(2^2 + 1)=5120\\$$
$$\\2^{n-2}(5)=5120\\$$
$$\\2^{n-2}=1024\\$$
$$\\2^{n-2}=2^{10}\\$$
$$\\n-2=10\\$$
$$\\n=12\\$$
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- EconomistGMATTutor
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Hi Max@Math Revolution,$$If\ \ 2^n+2^{n-2}=5120,\ then\ n=?$$
A. 8
B. 9
C. 10
D. 11
E. 12
Let's take a look at your question.
$$2^n+2^{n-2}=5120$$
$$2^n+2^n.2^{-2}=5120$$
Taking 2^n as a common factor:
$$2^n\left(1+2^{-2}\right)=5120$$
$$2^n\left(1+\frac{1}{2^2}\right)=5120$$
$$2^n\left(1+\frac{1}{4}\right)=5120$$
$$2^n\left(\frac{5}{4}\right)=5120$$
$$2^n=5120\times\frac{4}{5}$$
$$2^n=1024\times4$$
$$2^n=2^{10}\times2^2$$
$$2^n=2^{12}$$
$$n=12$$
Therefore, option E is correct.
Hope it helps.
I am available if you'd like any follow up.
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- Max@Math Revolution
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=>
Factoring yields
2^n+2^{n-2}=2^2^{2n-2}+2^{n-2}=(2^2+1)2^{n-2}=5*2^{n-2}=5120=5*1024.
Therefore,
2^{n-2}=1024=2^{10}
and
n-2 = 10.
It follows that
n = 12.
Therefore, the answer is E.
Answer : E
Factoring yields
2^n+2^{n-2}=2^2^{2n-2}+2^{n-2}=(2^2+1)2^{n-2}=5*2^{n-2}=5120=5*1024.
Therefore,
2^{n-2}=1024=2^{10}
and
n-2 = 10.
It follows that
n = 12.
Therefore, the answer is E.
Answer : E
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We can simplify the given equation:Max@Math Revolution wrote:[GMAT math practice question]
$$If\ \ 2^n+2^{n-2}=5120,\ then\ n=?$$
A. 8
B. 9
C. 10
D. 11
E. 12
2^n + 2^(n-2) = 5120
2^n + 2^n x 2^-2 = 5120
Factoring the common factor of 2^n from both terms on the left side of the equation, we have:
2^n(1 + 2^-2) = 5120
2^n(1 + 1/4) = 5120
2^n(5/4) = 5120
2^n = 5120 x 4/5
2^n = 4096
n = 12
Answer: E
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