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# 1/2^10 + 1/2^11 + 1/2^12 + 1/2^12

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Re: 1/2^10 + 1/2^11 + 1/2^12 + 1/2^12 [#permalink]
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given
$$\frac{1}{2^{10}} + \frac{1}{2^{11}} + \frac{1}{2^{12}} + \frac{1}{2^{12}}$$
can be written as
1/2^10 * ( 1+1/2+1/4+1/4 )
or say
1/2^10 * 8/4 ; 1/2^10 * 2 ;
1/2^9
OPTION C

Bunuel wrote:
$$\frac{1}{2^{10}} + \frac{1}{2^{11}} + \frac{1}{2^{12}} + \frac{1}{2^{12}}$$

A. $$\frac{1}{2^7}$$

B. $$\frac{1}{2^8}$$

C. $$\frac{1}{2^9}$$

D. $$\frac{1}{2^{13}}$$

E. $$\frac{1}{2^{45}}$$

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Re: 1/2^10 + 1/2^11 + 1/2^12 + 1/2^12 [#permalink]
Bunuel wrote:
$$\frac{1}{2^{10}} + \frac{1}{2^{11}} + \frac{1}{2^{12}} + \frac{1}{2^{12}}$$

A. $$\frac{1}{2^7}$$

B. $$\frac{1}{2^8}$$

C. $$\frac{1}{2^9}$$

D. $$\frac{1}{2^{13}}$$

E. $$\frac{1}{2^{45}}$$

First, let’s obtain the common denominator of 2^12. We obtain:

(2^2/2^2)(1/2^10) + (2/2)1/2^11) + 1/2^12 + 1/2^12

2^2/2^12 + 2/2^12 + 1/2^12 + 1/2^12 = 8/2^12 = 2^3/2^12 = 1/2^9

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Re: 1/2^10 + 1/2^11 + 1/2^12 + 1/2^12 [#permalink]
Top Contributor
Let try another approach:
The max value in the series is 1/2^10 and the rest of the terms are less than 1/2^10.

since there are 4 terms , we can say that SUM < 4 * 1/2^10
i.e. SUM < 1/2^8

The min value in the series is 1/2^12 and the rest of the terms are greater than 1/2^10.
SUM > 4 * 1/2^12
SUM > 1/2^10

We can conclude that 1/2^10 < SUM < 1/2^8

When we analyze the options, we can see that all options are in decreasing order and only 1/2^9 is in our range.

Option C is the right answer.

Thanks,
Clifin J Francis
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Re: 1/2^10 + 1/2^11 + 1/2^12 + 1/2^12 [#permalink]