Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized for You

we will pick new questions that match your level based on your Timer History

Track Your Progress

every week, we’ll send you an estimated GMAT score based on your performance

Practice Pays

we will pick new questions that match your level based on your Timer History

Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.

It appears that you are browsing the GMAT Club forum unregistered!

Signing up is free, quick, and confidential.
Join other 500,000 members and get the full benefits of GMAT Club

Registration gives you:

Tests

Take 11 tests and quizzes from GMAT Club and leading GMAT prep companies such as Manhattan GMAT,
Knewton, and others. All are free for GMAT Club members.

Applicant Stats

View detailed applicant stats such as GPA, GMAT score, work experience, location, application
status, and more

Books/Downloads

Download thousands of study notes,
question collections, GMAT Club’s
Grammar and Math books.
All are free!

Thank you for using the timer!
We noticed you are actually not timing your practice. Click the START button first next time you use the timer.
There are many benefits to timing your practice, including:

51% (01:13) correct 49% (00:53) wrong based on 103 sessions

HideShow timer Statistics

The value of \(\frac{1}{2} + (\frac{1}{2})^2 + (\frac{1}{2})^3 + ... + (\frac{1}{2})^{20}\) is between?

A. \(\frac{1}{2}\) and \(\frac{2}{3}\) B. \(\frac{2}{3}\) and \(\frac{3}{4}\) C. \(\frac{3}{4}\) and \(\frac{9}{10}\) D. \(\frac{9}{10}\) and \(\frac{10}{9}\) E. \(\frac{10}{9}\) and \(\frac{3}{2}\)

The value of \(\frac{1}{2} + (\frac{1}{2})^2 + (\frac{1}{2})^3 + ... + (\frac{1}{2})^{20}\) is between?

A. \(\frac{1}{2}\) and \(\frac{2}{3}\) B. \(\frac{2}{3}\) and \(\frac{3}{4}\) C. \(\frac{3}{4}\) and \(\frac{9}{10}\) D. \(\frac{9}{10}\) and \(\frac{10}{9}\) E. \(\frac{10}{9}\) and \(\frac{3}{2}\)

We have the sum of a geometric progression with the first term equal to \(\frac{1}{2}\) and the common ratio also equal to \(\frac{1}{2}\).

Now, the sum of infinite geometric progression with common ratio \(|r| \lt 1\), is \(sum=\frac{b}{1-r}\), where \(b\) is the first term. So, if we had infinite geometric progression instead of just 20 terms then its sum would be \(Sum=\frac{\frac{1}{2}}{1-\frac{1}{2}}=1\). Which means that the sum of this sequence will never exceed 1. Also since we have a large enough number of terms (20), the sum will be very close to 1, so we can safely choose answer choice D.

One can also use direct formula.

We have geometric progression with \(b=\frac{1}{2}\), \(r=\frac{1}{2}\) and \(n=20\);

\(S_n=\frac{b(1-r^n)}{(1-r)}\), so:

\(S_{20}=\frac{\frac{1}{2}(1-\frac{1}{2^{20}})}{(1-\frac{1}{2})}=1-\frac{1}{2^{20}}\). Since \(\frac{1}{2^{20}}\) is very small number then \(1-\frac{1}{2^{20}}\) will be less than 1 but very close to it.

In case we forget formula we can do it by following approximation 1/2=.50 1/2^2 =.250 1/2^3= .125 ............. .50+.250+.125 = .875 So we are sure that by other decimal number value we will be very close to 1 but less than 1

Only option D shows that

Last edited by Raihanuddin on 09 Nov 2015, 08:28, edited 1 time in total.

The whole issue with D was that 10/9 is 1.11... Which is greater than 1?

Please clarify

Note that the question asks about the range, not the exact value of the expression. If the value of the expression were 0.9999999 wouldn't it still be correct to say that it's between 0.9 and 1000000000000000000000000?
_________________

I actually forgot the formula when I had this question so I did the most native manual way I can: 1/2 + 1/4 = 3/4 + 1/8 = 7/8 + 1/16 = 15/16 Stop here: 15/16 > 9/10 and since the rhythm of the series we can see that the sum will always <1 --> correct ans: D _________________

[4.33] In the end, what would you gain from everlasting remembrance? Absolutely nothing. So what is left worth living for? This alone: justice in thought, goodness in action, speech that cannot deceive, and a disposition glad of whatever comes, welcoming it as necessary, as familiar, as flowing from the same source and fountain as yourself. (Marcus Aurelius)

I actually forgot the formula when I had this question so I did the most native manual way I can: 1/2 + 1/4 = 3/4 + 1/8 = 7/8 + 1/16 = 15/16 Stop here: 15/16 > 9/10 and since the rhythm of the series we can see that the sum will always <1 --> correct ans: D

I didn't even know the formula but took the same approach. I was able to solve the problem within 1 minute.

In case we forget formula we can do it by following approximation 1/2=.50 1/2^2 =.250 1/2^3= .125 ............. .50+.250+.125 = .875 So we are sure that by other decimal number value we will be very close to 1 but less than 1

Only option D shows that

To be sure, I added 1/2, 1/4,1/8 and 1/16 to to get 0.875 + 0.0625 > 0.93

and since the further we go, the smaller fractions we add, we should be close to 0.99 Hence D

I agree that option D is the closest in terms of the correct answer,but the range is wrong as 10/9 is greater than 1. So I had eliminated that option at first glance. I think none of the options in this question is correct.

I agree that option D is the closest in terms of the correct answer,but the range is wrong as 10/9 is greater than 1. So I had eliminated that option at first glance. I think none of the options in this question is correct.

Please re-read the solution and the discussion above. The value of the expression is between 9/10 and 10/9 (D). It's very slightly less than 1.
_________________

The whole issue with D was that 10/9 is 1.11... Which is greater than 1?

Please clarify

Note that the question asks about the range, not the exact value of the expression. If the value of the expression were 0.9999999 wouldn't it still be correct to say that it's between 0.9 and 1000000000000000000000000?

Bunuel, yes the values can be considered that way. But also we in this range there are values which are greater than 1 and we know for sure it cannot be greater than 1. How do we justify that...what if the value of the expression is 1.00000000009 or 1.10, wouldn't it be greater than 1? Pls clarify

The whole issue with D was that 10/9 is 1.11... Which is greater than 1?

Please clarify

Note that the question asks about the range, not the exact value of the expression. If the value of the expression were 0.9999999 wouldn't it still be correct to say that it's between 0.9 and 1000000000000000000000000?

Bunuel, yes the values can be considered that way. But also we in this range there are values which are greater than 1 and we know for sure it cannot be greater than 1. How do we justify that...what if the value of the expression is 1.00000000009 or 1.10, wouldn't it be greater than 1? Pls clarify

How does this matter?

Say x = 10. Wouldn't it be correct to say that it's between -1000000 and 10000000000000?
_________________

I actually forgot the formula when I had this question so I did the most native manual way I can: 1/2 + 1/4 = 3/4 + 1/8 = 7/8 + 1/16 = 15/16 Stop here: 15/16 > 9/10 and since the rhythm of the series we can see that the sum will always <1 --> correct ans: D

I didn't even know the formula but took the same approach. I was able to solve the problem within 1 minute.

Hi,

I understood that you did it without using the formula...but can u please explain how did u arrive ta the answer in details

I understood until point u get 15/16..after that how did u arrive at the answer?