In a certain sequence, each term after the first term is one-half the

Question

In a certain sequence, each term after the first term is one-half the previous term. If the tenth term of the sequence is between 0.0001 and 0.001, then the twelfth term of the sequence is between

(A) 0.0025 and 0.025
(B) 0.00025 and 0.0025
(C) 0.000025 and 0.00025
(D) 0.0000025 and 0.000025
(E) 0.00000025 and 0.0000025

Bunuel · Accepted Answer

In a certain sequence, each term after the first term is one-half the previous term. If the tenth term of the sequence is between 0.0001 and 0.001, then the twelfth term of the sequence is between

(A) 0.0025 and 0.025
(B) 0.00025 and 0.0025
(C) 0.000025 and 0.00025
(D) 0.0000025 and 0.000025
(E) 0.00000025 and 0.0000025

Each term after the first term is one-half the previous term:  a_{n+1}=\frac{a_n}{2} . So:

a_{11}=\frac{a_{10}}{2}

a_{12}=\frac{a_{11}}{2}=\frac{a_{10}}{4}

4a_{12}=a_{10} .

Given:  0.0001 < a_{10}< 0.001 ;

0.0001 < 4a_{12}< 0.001 ;

0.000025 < a_{12}< 0.00025 .

Answer: C.

adiagr · Answer

12th Term will be obtained by multiplying the 10th term by  \frac{1}{2}  *  \frac{1}{2}

Let 10th term is a

0.0001 <  a_{10}  < 0.001

12th term will be

\frac{0.0001}{4}  <  a_{12}  <  \frac{0.001}{4}

0.000025 <  a_{12}  < 0.00025

Option C is the answer

damybox · Answer

the low limit of the number is equal to 1/10 of the highter limit of the number.

if you divide the low limit you'll find 0.000025

as a consequence the high limit of the number will be the low one *10 = 0.00025

answer C

hero_with_1000_faces · Answer

Multiply original number 0.0001 and 0.001 with 1,00,000 both, you get:

10 and 100, divide both by 2, 5 and 50 further divide by 2, 2.5 and 25, now divide by 1,00,000 or push both numbers by five zeros, you get "C"

hero_with_1000_faces · Answer

Multiply original number 0.0001 and 0.001 with 1,00,000 both, you get:

10 and 100, divide both by 2, you get 5 and 50,
further divide 5 and 50, by 2, you get 2.5 and 25,

now divide by 1,00,000 or push both numbers by five zeros, you get "C". Tadaaaaaaaaaaaa!

Jsound996 · Answer

Convert 0.0001 and 0.001 into scientific notation
 0.0001 = 1* 10^-4 
 0.001 = 1* 10^-3

Because each term is 1/2 of the previous term,  Term 12 = \frac{1}{2}*\frac{1}{2}*Term 10

Term 12 = \frac{1}{4}*Term 10

(.25* 10^-4) =  \frac{1}{4} *(1* 10^-4)

(.25* 10^-3) =  \frac{1}{4} *(1* 10^-3)

Then move the decimal place 4 and 3 spots to the left, which gives you  0.000025   and  0.00025

Answer is C

energetics · Answer

\frac{1}{10000} < n_{10} < \frac{1}{1000}

(\frac{1}{2^{2}}) * \frac{1}{10000} < n_{12} < (\frac{1}{2^{2}})* \frac{1}{1000}

\frac{1}{4} * 10^{-4} < n_{12} < \frac{1}{4} * 10^{-3}

.25 * 10^{-4} < n_{12} < .25 * 10^{-3}

.000025 < n_{12} < .00025

medinib · Answer

In a certain sequence, each term after the first term is one-half the previous term. If the tenth term of the sequence is between 0.0001 and 0.001, then the twelfth term of the sequence is between

(A) 0.0025 and 0.025
(B) 0.00025 and 0.0025
(C) 0.000025 and 0.00025
(D) 0.0000025 and 0.000025
(E) 0.00000025 and 0.0000025

12th term would be 10th term divided by 4 as each consecutive term is one half of the previous term.

a10/4 = 0.0001/4

1/10000*4

1/10000 * 1/4

1/10000 * 0.25

For four 0's move the decimal four places to the left.

.000025

Only option (C) matches

hero_with_1000_faces · Answer

Alternate solution, 12th term means: you need to divide the number twice by 2 i.e by 4 (1/2*1/2).

when you divide 0.001/4 you get 0.00025; easiest way.

kokus · Answer

Is the following approach correct:

Since the tenth term has 3 to 4 values after the decimal point, the eleventh term will have 4 to 5 values and the 12th will have 5 to 6. Since 25 takes two places, the zeroes in front will be either three or four as in answer C?

AnirudhaS · Answer

risky to assume. I will do it this way -

10th term: 0.0001 TO 0.001
11th term: 0.00005 TO 0.0005 (divide the above by 2)
12th term: 0.000025 TO 0.00025 (divide the above by 2)

kokus · Answer

Thanks, but why is it risky? You just need to be aware what the product will be and then just add the number of zeroes?

Kinshook · Answer

Given: In a certain sequence, each term after the first term is one-half the previous term.
Asked: If the tenth term of the sequence is between 0.0001 and 0.001, then the twelfth term of the sequence is between

Let the first term of the sequence be x
 T_1 = x 
 T_2 = \frac{x}{2} 
 T_n = \frac{x}{2^{n-1}}

T_{10} = \frac{x}{2^9} ; 
 .0001 < \frac{x}{2^9} < .001

T_{12} = \frac{x}{2^{11}} 
 \frac{.0001}{4} < T_{12} < \frac{.001}{4} 
 .000025 < T_{12} < .00025

IMO C

Basshead · Answer

10th term = between 0.0001 and 0.001

Convert to fraction:  \frac{1}{10000}  and  \frac{1}{1000}

11th term = between  \frac{1}{20000}  and  \frac{1}{2000}

12th term = between  \frac{1}{40000}  and  \frac{1}{4000}

12th term = between  \frac{1}{4} * \frac{1}{10000}  and  \frac{1}{4} * \frac{1}{1000}

12th term = between  0.000025  and  0.00025

Sabby · Answer

Instead of 0.0001 and 0.001, get rid of the decimals by multiplying the two numbers by 10^4 which gives us 1 and 10 (we'll put the decimals back later by multiplying the results by 10^-4)

1*1/2= 0,5 and 10*1/2=5
0,5*1/2=0,25 and 5*1/2=2,5

Result is : 0,25*10^-4= 0,000025 and 2,5*10^-4 = 0,00025

Answer C)

CEdward · Answer

Can we just say that the exponent is the number of zeros between the decimal and the first non-zero digit?

Sabby · Answer

Hi  CEdward ,

Not really.

Here are a few examples :
125/1000= 125*10^-3 = 0,125 Here there are no zeros

12/10000 = 12*10^-4 = 0,0012 Here there are 2 zeros between the decimal and the first non zero integer, and the exponent is -4

CEdward · Answer

Thanks! What's the fastest way to do such divisions without long division?

Posted from my mobile device

Sabby · Answer

CEdward ,

Actually you don't need to perform a division, you just need to move the decimal to the left and the exponent gives you the number of moves.

When you have 12/10000 or 12*10^-4 for example, it means that you have 4 digits after the decimal. You already have your two digits 1 and 2 therefore you just need to add two zeros. It gives you 0,0012

For example 12/10 or 12*10^-1 means you have 1 digit after the decimal therefore that digit is 2 and it gives you 1.2 
OR you can think of it as moving the decimal one place to the left

When you have for example 0.25*10^-2 it means you have to add two zeros between your decimal and the digit 2 . It is also like having 25*10^-4 and it means 4 digits after the decimal ==> 0.0025

Another example : 0.125 *10^-3 means you have to add 3 zeros between your decimal and the digit 1 ==> 0.000125

Does this help?

I also like to call that "number hopping" where the exponent gives you the number of hops or jumps you need to perform.
I drew it in the attached image.

In a certain sequence, each term after the first term is one-half the

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Problem Solving (PS) Questions

In a certain sequence, each term after the first term is one-half the

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards