In the sequence above each term after

Question

1/2, 1/4, 1/8, 1/16, 1/32, ....

In the sequence above each term after after the first one-half the previous term. If x is the tenth term of the sequence, then x satisfies which of the following inequalities?

A) 0.1 < x < 1
B) 0.01 < x < 0.1
C) 0.001 < x < 0.01
D) 0.0001 < x < 0.001
E) 0.00001 < x < 0.0001

shashanksst94 · Accepted Answer

In the Sequence notice that the sequence is just the 1/(2^n) ...
so for 1st term=1/2^1=1/2
 2nd term=1/(2^2)1/4, 3rd term=1/(2^3)=1/8 and so on...
Thus the 10th term will be 1/(2^10)=1/1024
Roughly, 1/1024 can be 1/1000=0.001 but since denominator is a bit more than 1000 therefore the actual value will be a bit less than 0.001.

thus the ans will lie btw. 0.0001 and 0.001.(D)

BrentGMATPrepNow · Answer

We can think of the terms as follows....
term_1 = 1/2
term_2 = (1/2)(1/2) = (1/2)²
term_3 = (1/2)(1/2)(1/2) = (1/2)³
term_4 = (1/2)(1/2)(1/2)(1/2) = (1/2)⁴
.
.
.
term_n = (1/2)(1/2)(1/2)(1/2) = (1/2)^n

So, term_10 = (1/2)¹⁰ = 1/2¹⁰ = 1/1,024

We get: 1/10,000 < 1/1,024 < 1/1,000
In other words: 0.0001 < 1/1,024 < 0.01

Answer: D

msk0657 · Answer

Hi,

This is how I tried.

We are asked to find the tenth term of the series 1/2, 1/4, 1/8, 1/16, 1/32, ....

If you observe that denominator is power of 2 i.e.  2^1 ,  2^2 , 2^3 .... then tenth term will be 2^10.

Here 2^10 = 1024

Then 1/1024 = 0.024. Now check where this number fits in and this value is greater than 0.001. So C is eliminated.

Then only D satisfies this condition.

syahasa2 · Answer

It's just about calculating the tenth term which is 2^10=1024.
Then you should be able to define where this number fits.

drbyrnes · Answer

What is your approach to solve for 2^10 and 1/1024?

generis · Answer

Then 1/1024 = 0.024 
 msk0657  , how did you get this value?

On a calculator it comes out as .0009765625.

ocelot22 · Answer

When comparing powers of some value to either powers of 10 or some decimal with a trailing 1 (which is also a power of 10), one approach is to rewrite our term, to get it as close to a power of 10 as possible. For (1/2)^10, a base of 1/8 is gonna be as close to 1/10 as we can get. Then (1/2)^10 = 1/(2^3 *2^3 *2^3 *2) = (1/8)^3*1/2 is approximately 
 (1/10)^3 *1/2 = .001*1/2 = approximately .0005, which falls between .0001 and .001. The answer is D

jfranciscocuencag · Answer

Hello msk0657 !

How did you get this result?

Then 1/1024 = 0.024

Kind regards!

sanjna2023 · Answer

We calculate 2^10 = 1024

Thus,

1/1000 > 1/1024
0.001 > 1/1024

Only option D fits with this calculation so we can easily select the same.

sarthak1701 · Answer

10th term will be 1/2^10 = 1/1024, this is roughly equal to 0.001 which satisfies the D inequality

Dash93541 · Answer

This is how i tried the question:

Since, a3 = 1/8 = 0.125
 a4 = 1/16 = 0.0625
 a5 = 1/32 = 0.03125

This showed a pattern that the 0s after the decimal will increase after every 3 intervals i.e for a6 = 1/64 = there will be 1 Zero after the decimal and it will change from a7 (2 Zeros continued till a9) then will change again at a10 where there will be 3 zeros.

Thus, we have the answer D

jaykayes · Answer

Knowing that it's a GMAT question, there MUST be a way to solve it without calculating the value of 2^10.
Here’s one way it can be done:
 Since all the answers are ranges, it's clear we only need to approximate the value of the tenth term.
 Use the properties of exponents by splitting 2^10 into 2^5 x 2^5.
 We already know that the 5th term is 1/2^5, i.e. 1/32.
 1/32 is slightly smaller than 1/30
 1/30 x 1/30 = 1/900
 Since all the answers are in decimals, convert this to a decimal.
 We know that 1/9 = 0.11111111111....
 So 1/900 = 0.001111111111....
 And the tenth term must be a bit smaller than this.
 We can eliminate options A, B, C.
 It’s also clear that the value of the tenth term can’t be as low as one-tenth of this
 So it must definitely be greater than 0.0001 - that's the upper limit of option E, so eliminate it.
 We're left with the correct answer choice D

In the sequence above each term after

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