If 2^n+2^n-2=5120, then 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

We can PLUG IN THE ANSWERS, which represent the value of n.
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:

$$\\2^n+2^{n-2}=5120\\$$
$$\\2^{n-2}(2^2 + 1)=5120\\$$
$$\\2^{n-2}(5)=5120\\$$
$$\\2^{n-2}=1024\\$$
$$\\2^{n-2}=2^{10}\\$$
$$\\n-2=10\\$$
$$\\n=12\\$$

$$If\ \ 2^n+2^{n-2}=5120,\ then\ n=?$$

A. 8
B. 9
C. 10
D. 11
E. 12

Hi Max@Math Revolution,
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.

=>
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