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For every integer k from 1 to 10, inclusive, the kth term of

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Re: Numbers , Squences , Indices [#permalink]  23 Apr 2013, 01:01
Expert's post
kabilank87 wrote:
For every integers K from 1 to 10 inclusive, the K th term of a certail sequence is given by [(-1)^(K+1)](1 / (2^K). What is the sum of first 10 terms of the sequence ?

a. Greater than 2
b. Between 1 and 2
c. Between 1/2 and 1
d.Between 1/4 and 1/2
e.Less than 1/4

$$S = (\frac{1}{2}+\frac{1}{2^{3}}+..\frac{1}{2^{9}}) - (\frac{1}{2^{2}}+..\frac{1}{2^{10}})$$

Multiply on both sides by 2:

$$2S = 1+(\frac{1}{2^{2}}+..\frac{1}{2^{8}})-(\frac{1}{2}+\frac{1}{2^{3}}+..\frac{1}{2^{9}})$$

$$3S = 1-\frac{1}{2^{10}}$$

$$S = \frac{1}{3}-\frac{1}{3*2^{10}}$$

D.
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Last edited by mau5 on 24 Apr 2013, 11:54, edited 2 times in total.
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Re: Numbers , Squences , Indices [#permalink]  23 Apr 2013, 02:57
Zarrolou wrote:
kabilank87 wrote:
For every integers K from 1 to 10 inclusive, the K th term of a certail sequence is given by [(-1)^(K+1)](1 / (2^K). What is the sum of first 10 terms of the sequence ?

a. Greater than 2
b. Between 1 and 2
c. Between 1/2 and 1
d.Between 1/4 and 1/2
e.Less than 1/4

I answer this question in my second attempt, but it takes a lot of times around 4 minutes. Please explain a shorter way to do this ?

Here we need to find a pattern
$$\frac{1}{2},-\frac{1}{4},\frac{1}{8},...$$ as you see the sign changes every term.
The first and bigger is 0,5 and then we subtract and sum smaller and smaller terms.
We can eliminate any option that gives us a upper limit greater than 1/2.
We are down to D and E. Is the sum less than 1/4?
Take the sum of pair of terms : the first 2 give us 1/4, the second pair is 1/8-1/16 positive so we add value to 1/4, so the sum will be greater.(this is true also for the next pairs, so we add to 1/ 4 a positive value for each pair)
D

Hope its clear, let me know

Hi Zarro ,

Very clear. But how to approach this kind of problems in GMAT without taking much time. Even though you explanation look very simple and time saving , i am not sure how i respond to the question same way as you explained. Do you have any siggestions how can i approach these problems ?

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Re: Numbers , Squences , Indices [#permalink]  23 Apr 2013, 03:11
A quick way to do this :

the sum of the sequence for k from 1 to 10 is : $$Sum = 1/2 - 1/(2^2) + 1/(2^3) ....$$

Notice that ... represents a very small numbers that even substracted or added to the first three terms (1/2 , -1/4 and 1/8), we can neglect it .

Hence, since 1/2-1/4+1/8 is between 1/4 and 1/2 , the sum will be between 1/4 and 1/2 as well.

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Re: Numbers , Squences , Indices [#permalink]  23 Apr 2013, 03:13
kabilank87 wrote:

Hi Zarro ,

Very clear. But how to approach this kind of problems in GMAT without taking much time. Even though you explanation look very simple and time saving , i am not sure how i respond to the question same way as you explained. Do you have any siggestions how can i approach these problems ?

Hi kabilank87,

when we deal with a series (as in this case) the first and most important thing to do is find a pattern.

One you've found that you can continue the series with no limit ( the GMAT will never ask you the exact value of a seires such this one), but the role of patterns is crucial also in question where you're asked to find the $$N^t^h$$ term of a sequence.
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Re: Numbers , Squences , Indices [#permalink]  24 Apr 2013, 11:21
Zarrolou wrote:
kabilank87 wrote:
For every integers K from 1 to 10 inclusive, the K th term of a certail sequence is given by [(-1)^(K+1)](1 / (2^K). What is the sum of first 10 terms of the sequence ?

a. Greater than 2
b. Between 1 and 2
c. Between 1/2 and 1
d.Between 1/4 and 1/2
e.Less than 1/4

I answer this question in my second attempt, but it takes a lot of times around 4 minutes. Please explain a shorter way to do this ?

Here we need to find a pattern
$$\frac{1}{2},-\frac{1}{4},\frac{1}{8},...$$ as you see the sign changes every term.
The first and bigger is 0,5 and then we subtract and sum smaller and smaller terms.
We can eliminate any option that gives us a upper limit greater than 1/2.
We are down to D and E. Is the sum less than 1/4?
Take the sum of pair of terms : the first 2 give us $$\frac{1}{4}$$, the second pair is $$\frac{1}{8}-\frac{1}{16}$$ positive so we add value to $$\frac{1}{4}$$, so the sum will be greater.(this is true also for the next pairs, so we add to $$\frac{1}{4}$$ a positive value for each pair)
D

Hope its clear, let me know

Can you please elaborate more , Zarrou ? I still do not understand , and the line in red tricks me a lot .

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Re: Numbers , Squences , Indices [#permalink]  24 Apr 2013, 11:37
1
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TheNona wrote:

Can you please elaborate more , Zarrou ? I still do not understand , and the line in red tricks me a lot .

The pattern:
$$\frac{1}{2},-\frac{1}{4},\frac{1}{8},-\frac{1}{16},...$$

We have to sum those elements so:
$$\frac{1}{2}-\frac{1}{4}+\frac{1}{8}-\frac{1}{16}+...$$
The first term is $$\frac{1}{2}$$, to this we subtract 1/4, to the result we add 1/8, and so on
As you see the operations involve smaller and smaller term each time. The first thing to notice here is that the sum will be <1/2, we can easily see this:
$$\frac{1}{2}-\frac{1}{4}=\frac{1}{4}$$ and the operations will not produce a result >1/2. Hope it's clear here: the numbers decrease too rapidly to produce a result as big as the first term!

Now we are left with D and E: the only 2 option which result is <1/2. And the question is: will the sum be less than 1/4?
We have to find an easy way to see this, consider this fact:
$$\frac{1}{2},-\frac{1}{4},\frac{1}{8},-\frac{1}{16},...$$
take the sum of couple of terms: 1st with 2nd, 3rd with 4th, and so on...
The result will be positive for each couple, lets take a look:$$\frac{1}{2}-\frac{1}{4}=\frac{1}{4}$$ for the first one, $$+\frac{1}{8}-\frac{1}{16}=\frac{1}{16}(>0)$$ and so on.

The thing to take away here is: 1/4+(num>0)+(num>0)+... will NOT be less than 1/4, how could it be if all numbers are positive?

So the sum will be GREATER than 1/4 and LESSER than 1/4.

Hope everything is clear now, I have been as exhaustive as possible, let me know
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Re: Numbers , Squences , Indices [#permalink]  24 Apr 2013, 11:47
Zarrolou wrote:
TheNona wrote:

Can you please elaborate more , Zarrou ? I still do not understand , and the line in red tricks me a lot .

The pattern:
$$\frac{1}{2},-\frac{1}{4},\frac{1}{8},-\frac{1}{16},...$$

We have to sum those elements so:
$$\frac{1}{2}-\frac{1}{4}+\frac{1}{8}-\frac{1}{16}+...$$
The first term is $$\frac{1}{2}$$, to this we subtract 1/4, to the result we add 1/8, and so on
As you see the operations involve smaller and smaller term each time. The first thing to notice here is that the sum will be <1/2, we can easily see this:
$$\frac{1}{2}-\frac{1}{4}=\frac{1}{4}$$ and the operations will not produce a result >1/2. Hope it's clear here: the numbers decrease too rapidly to produce a result as big as the first term!

Now we are left with D and E: the only 2 option which result is <1/2. And the question is: will the sum be less than 1/4?
We have to find an easy way to see this, consider this fact:
$$\frac{1}{2},-\frac{1}{4},\frac{1}{8},-\frac{1}{16},...$$
take the sum of couple of terms: 1st with 2nd, 3rd with 4th, and so on...
The result will be positive for each couple, lets take a look:$$\frac{1}{2}-\frac{1}{4}=\frac{1}{4}$$ for the first one, $$+\frac{1}{8}-\frac{1}{16}=\frac{1}{16}(>0)$$ and so on.

The thing to take away here is: 1/4+(num>0)+(num>0)+... will NOT be less than 1/4, how could it be if all numbers are positive?

So the sum will be GREATER than 1/4 and LESSER than 1/4.

Hope everything is clear now, I have been as exhaustive as possible, let me know

Perfect! Thanks a lot
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Re: Numbers , Squences , Indices [#permalink]  24 Apr 2013, 23:55
1
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Expert's post
TheNona wrote:
Zarrolou wrote:
kabilank87 wrote:
For every integers K from 1 to 10 inclusive, the K th term of a certail sequence is given by [(-1)^(K+1)](1 / (2^K). What is the sum of first 10 terms of the sequence ?

a. Greater than 2
b. Between 1 and 2
c. Between 1/2 and 1
d.Between 1/4 and 1/2
e.Less than 1/4

I answer this question in my second attempt, but it takes a lot of times around 4 minutes. Please explain a shorter way to do this ?

Here we need to find a pattern
$$\frac{1}{2},-\frac{1}{4},\frac{1}{8},...$$ as you see the sign changes every term.
The first and bigger is 0,5 and then we subtract and sum smaller and smaller terms.
We can eliminate any option that gives us a upper limit greater than 1/2.
We are down to D and E. Is the sum less than 1/4?
Take the sum of pair of terms : the first 2 give us $$\frac{1}{4}$$, the second pair is $$\frac{1}{8}-\frac{1}{16}$$ positive so we add value to $$\frac{1}{4}$$, so the sum will be greater.(this is true also for the next pairs, so we add to $$\frac{1}{4}$$ a positive value for each pair)
D

Hope its clear, let me know

Can you please elaborate more , Zarrou ? I still do not understand , and the line in red tricks me a lot .

Check out the GP perspective on this question too. It really cuts down your work:
http://www.veritasprep.com/blog/2012/04 ... rspective/
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Get started with Veritas Prep GMAT On Demand for $199 Veritas Prep Reviews Manager Joined: 07 Apr 2012 Posts: 126 Location: United States Concentration: Entrepreneurship, Operations Schools: ISB '15 GMAT 1: 590 Q48 V23 GPA: 3.9 WE: Operations (Manufacturing) Followers: 0 Kudos [?]: 9 [0], given: 45 Re: For every integer k from 1 to 10, inclusive, the kth term of [#permalink] 11 Oct 2013, 12:22 Since I solved it from a different method than mentioned here, thought it to share. I guess we all might have deudced there are 10 terms and alternately positive and negative. I tried with GP sum formula and got lost in calculation. Since we have alternately + - we can make use of it. Take 1st term , 2nd term 1/2 and -1/4, add them to get 1/4 Similarly 3rd and 4rth term gives you 1/8 ( 1/8 + -1/16) 1/16 We see a multiplication pattern of 4 here so no need to calculate further. 1/4, 1/16, 1/64, 1/256, 1/1024 Add them to get 256+64+16+4+1= 341/1024 Clealry less than half so lies b/w 1/4 and 1/2 Not a very great method but I guess helps me avoid calculcation mistake if I go for GP sum. Manager Joined: 18 Oct 2013 Posts: 81 Location: India Concentration: Technology, Finance GMAT 1: 580 Q48 V21 GMAT 2: 530 Q49 V13 GMAT 3: 590 Q49 V21 WE: Information Technology (Computer Software) Followers: 3 Kudos [?]: 13 [0], given: 36 Re: For every integer k from 1 to 10, inclusive, the kth term of [#permalink] 28 Feb 2014, 22:18 Question is written in such a way that it is difficult to comprehend but once it is then it is just a matter of few seconds to crack.Here my answer. T(K) = (−1)^k+1 * 1/2^k So, T(1) =1/2 T(2)= -1/4 T(3)= 1/8 T(4)= -1/16 .... and so on No need to calculate higher terms because they doesn't produce any significant increase in sum as answers are widely distributed. Sum=(1/2) + (-1/4) +(1/8) +(-1/16)+..... =>1/4 +1/16 => 0.25+0.0625 => .3125 SO answer is between 0.25 and 0.5 . option D Option A,B & E can be rejected just by looking terms as they are too big or small enough. Proper calculation approach: even though u calculate each term Sum=1/4+1/16+1/64+1/256+1/1024 =>0.25+0.0625+0.015625+0.00390625 +0.000976 =0.3330 Intern Joined: 12 Nov 2013 Posts: 2 Followers: 0 Kudos [?]: 0 [0], given: 0 Exponents / powers (2) [#permalink] 13 Mar 2014, 07:17 How can I approach this question? (-1)k+1(½k). T is the sum of the first 10 k, is t a. > 2 b. between 1 and 2 c. between ½ and 1 d. between ¼ and ½ e. < ¼ Math Expert Joined: 02 Sep 2009 Posts: 30437 Followers: 5102 Kudos [?]: 57613 [0], given: 8819 Re: Exponents / powers (2) [#permalink] 13 Mar 2014, 07:42 Expert's post chrish06 wrote: How can I approach this question? (-1)k+1(½k). T is the sum of the first 10 k, is t a. > 2 b. between 1 and 2 c. between ½ and 1 d. between ¼ and ½ e. < ¼ Merging similar topics. please refer to the discussion above. _________________ Manager Joined: 11 Sep 2013 Posts: 152 Concentration: Finance, Finance Followers: 1 Kudos [?]: 50 [0], given: 154 Re: For every integer k from 1 to 10, inclusive, the kth term of [#permalink] 11 Apr 2014, 23:02 Sequence will be 1/2-1/4+1/8-1/16....... Sum of first pair= 1/4 = 25% Sum of 2nd pair=1/16 = Appropriately 6% So we can see that % is reducing. Therefore next three percentage each of which will not be more than 6% So, Sum= 25%+6%+ ........ = in between (25 and 50)% D shows that in percentage. Intern Joined: 24 Jun 2014 Posts: 48 Followers: 0 Kudos [?]: 16 [0], given: 187 Re: For every integer k from 1 to 10, inclusive, the kth term of [#permalink] 05 Nov 2014, 21:58 Bunuel wrote: For every integer k from 1 to 10, inclusive the "k"th term of a certain sequence is given by $$(-1)^{(k+1)}*(\frac{1}{2^k})$$ if T is the sum of the first 10 terms in the sequence, then T is A. Greater than 2 B. Between 1 and 2 C. Between 1/2 and 1 D. Between 1/4 and 1/2 E. Less than 1/4 First of all we see that there is set of 10 numbers and every even term is negative. Second it's not hard to get this numbers: $$\frac{1}{2}$$, $$-\frac{1}{4}$$, $$\frac{1}{8}$$, $$-\frac{1}{16}$$, $$\frac{1}{32}$$... enough for calculations, we see pattern now. And now the main part: adding them up is quite a job, after calculations you'll get $$\frac{341}{1024}$$. You can add them up by pairs but it's also time consuming. Once we've done it we can conclude that it's more than $$\frac{1}{4}$$ and less than $$\frac{1}{2}$$, so answer is D. BUT there is shortcut: Sequence $$\frac{1}{2}$$, $$-\frac{1}{4}$$, $$\frac{1}{8}$$, $$-\frac{1}{16}$$, $$\frac{1}{32}$$... represents geometric progression with first term $$\frac{1}{2}$$ and the common ratio of $$-\frac{1}{2}$$. Now, the sum of infinite geometric progression with common ratio |r|<1[/m], is $$sum=\frac{b}{1-r}$$, where $$b$$ is the first term. So, if the sequence were infinite then the sum would be: $$\frac{\frac{1}{2}}{1-(-\frac{1}{2})}=\frac{1}{3}$$ This means that no matter how many number (terms) we have their sum will never be more then $$\frac{1}{3}$$ (A, B and C are out). Also this means that the sum of our sequence is very close to $$\frac{1}{3}$$ and for sure more than $$\frac{1}{4}$$ (E out). So the answer is D. Answer: D. Other solutions at: sequence-can-anyone-help-with-this-question-88628.html#p668661 Alternative, if you use the geometric series formula. S = \frac{a(1-r^n)}{1-r} where a = first term, r = multiple factor, n = # of terms. Hi Bunuel, how are these two formula different? Thank you. Math Expert Joined: 02 Sep 2009 Posts: 30437 Followers: 5102 Kudos [?]: 57613 [0], given: 8819 Re: For every integer k from 1 to 10, inclusive, the kth term of [#permalink] 06 Nov 2014, 05:46 Expert's post vietnammba wrote: Bunuel wrote: For every integer k from 1 to 10, inclusive the "k"th term of a certain sequence is given by $$(-1)^{(k+1)}*(\frac{1}{2^k})$$ if T is the sum of the first 10 terms in the sequence, then T is A. Greater than 2 B. Between 1 and 2 C. Between 1/2 and 1 D. Between 1/4 and 1/2 E. Less than 1/4 First of all we see that there is set of 10 numbers and every even term is negative. Second it's not hard to get this numbers: $$\frac{1}{2}$$, $$-\frac{1}{4}$$, $$\frac{1}{8}$$, $$-\frac{1}{16}$$, $$\frac{1}{32}$$... enough for calculations, we see pattern now. And now the main part: adding them up is quite a job, after calculations you'll get $$\frac{341}{1024}$$. You can add them up by pairs but it's also time consuming. Once we've done it we can conclude that it's more than $$\frac{1}{4}$$ and less than $$\frac{1}{2}$$, so answer is D. BUT there is shortcut: Sequence $$\frac{1}{2}$$, $$-\frac{1}{4}$$, $$\frac{1}{8}$$, $$-\frac{1}{16}$$, $$\frac{1}{32}$$... represents geometric progression with first term $$\frac{1}{2}$$ and the common ratio of $$-\frac{1}{2}$$. Now, the sum of infinite geometric progression with common ratio |r|<1[/m], is $$sum=\frac{b}{1-r}$$, where $$b$$ is the first term. So, if the sequence were infinite then the sum would be: $$\frac{\frac{1}{2}}{1-(-\frac{1}{2})}=\frac{1}{3}$$ This means that no matter how many number (terms) we have their sum will never be more then $$\frac{1}{3}$$ (A, B and C are out). Also this means that the sum of our sequence is very close to $$\frac{1}{3}$$ and for sure more than $$\frac{1}{4}$$ (E out). So the answer is D. Answer: D. Other solutions at: sequence-can-anyone-help-with-this-question-88628.html#p668661 Alternative, if you use the geometric series formula. S = \frac{a(1-r^n)}{1-r} where a = first term, r = multiple factor, n = # of terms. Hi Bunuel, how are these two formula different? Thank you. The formula I used is for the sum of infinite geometric progression with common ratio |r|<1. _________________ EMPOWERgmat Instructor Status: GMAT Assassin/Co-Founder Affiliations: EMPOWERgmat Joined: 19 Dec 2014 Posts: 4680 Location: United States (CA) GMAT 1: 800 Q51 V49 GRE 1: 340 Q170 V170 Followers: 196 Kudos [?]: 1307 [0], given: 139 For every integer k from 1 to 10, inclusive, the kth term of a c [#permalink] 20 Apr 2015, 17:42 Expert's post Hi All, As complex as this question looks, it's got a great pattern-matching 'shortcut' built into it. When combined with the answer choices, you can avoid some of the calculations.... By plugging in the first few numbers (1, 2, 3, 4), you can see that a pattern emerges among the terms.... 1st term = 1/2 2nd term = -1/4 3rd term = 1/8 4th term = -1/16 The terms follow a positive-negative-positive-negative pattern all the way to the 10th term and each term is the product of the prior term and 1/2. By "pairing up' the terms, another pattern emerges.... 1/2 - 1/4 = 1/4 1/8 - 1/16 = 1/16 1/32 - 1/64 = 1/64 Etc. The "pairs" get progressively smaller (notice how each is the product of the prior term and 1/4). This means that we're "starting with" 1/4 and adding progressively TINIER fractions to it. Since we're just adding 4 progressively smaller fractions to 1/4, this means that we're going to end up with a total that's just a LITTLE MORE than 1/4. Looking at the answer choices, there's only one answer that fits: Final Answer: [Reveal] Spoiler: D GMAT assassins aren't born, they're made, Rich _________________ Intern Joined: 14 Oct 2013 Posts: 41 GMAT 1: Q V Followers: 0 Kudos [?]: 3 [0], given: 118 Re: For every integer k from 1 to 10, inclusive, the kth term of [#permalink] 10 May 2015, 18:00 I've never seen this term "geometric progression" in my studies thus far - is there a good overview of them somewhere and potential questions that might be asked in reference to them? Thanks! Veritas Prep GMAT Instructor Joined: 16 Oct 2010 Posts: 6062 Location: Pune, India Followers: 1605 Kudos [?]: 8971 [2] , given: 195 Re: For every integer k from 1 to 10, inclusive, the kth term of [#permalink] 10 May 2015, 20:07 2 This post received KUDOS Expert's post healthjunkie wrote: I've never seen this term "geometric progression" in my studies thus far - is there a good overview of them somewhere and potential questions that might be asked in reference to them? Thanks! Here is a post that explains Geometric progressions (GP): http://www.veritasprep.com/blog/2012/04 ... gressions/ The GP perspective on this question is discussed here: http://www.veritasprep.com/blog/2012/04 ... rspective/ _________________ Karishma Veritas Prep | GMAT Instructor My Blog Get started with Veritas Prep GMAT On Demand for$199

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Re: For every integer k from 1 to 10, inclusive, the kth term of [#permalink]  11 May 2015, 10:17
Expert's post
Hi healthjunkie,

Geometric progressions are rather rare on the GMAT (while you will see at least 1 sequence question on Test Day, it is not likely to be a Geometric sequence), so you shouldn't be putting too much effort into this concept just yet.

How are you performing on the Quant section overall? How about in the 'big' categories (Algebra, Arithmetic, Number Properties, DS, etc.)? That's where you're going to find the bulk of the points.

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Re: For every integer k from 1 to 10, inclusive, the kth term of [#permalink]  12 Oct 2015, 20:33
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VeritasPrepKarishma wrote:
prathns wrote:
For every integer k from 1 to 10, inclusive, the kth term of a certain sequence is given by (-1)^(k+1) *(1/2^k). If T is the sum of the first 10 terms in the sequence then T is

a)greater than 2
b)between 1 and 2
c)between 1/2 and 1
d)between 1/4 and 1/2
e)less than 1/4.

I have no clue what info has been given and how to use it to derive T.

Kindly post a detailed explanation.

Thanks.
Prath.

Using some keen observation, you can quickly arrive at the answer...
Terms will be: $$\frac{1}{2} - \frac{1}{4} + \frac{1}{8} - \frac{1}{16} + \frac{1}{32} - ... - \frac{1}{1024}$$
For every pair of values:
$$\frac{1}{2} - \frac{1}{4} = \frac{1}{4}$$

$$\frac{1}{8} - \frac{1}{16} = \frac{1}{16}$$
etc...

So this series is actually just
$$\frac{1}{4} + \frac{1}{16} + ... + \frac{1}{1024}$$

So the sum is definitely greater than 1/4.
When you add an infinite GP with 1/16 as first term and 1/4 as common ratio, the sum will be $$\frac{\frac{1}{16}}{1-\frac{1}{4}} = 1/12$$. Here, the sum of terms 1/16 + 1/64 + ... 1/1024 is definitely less than 1/12. Hence the sum is definitely less than 1/2. Answer is (D).

Quote:
Hi Karishma

The first term in this example is 1/2. Can you kindly explain how to calculate the sum of all terms of a GP with constant ratio >1 ?

Thanks

Sum of n terms of a GP = a(1 - r^n)/(1 - r)

The formula is the same whether |r| is more than 1 or less than 1.

You can find the sum of an infinite GP by the formula a/(1 - r) only when |r| < 1.
You cannot find the sum of an infinite GP when |r| > 1 because the sum will be infinite.
e.g. 3 + 9 + 27 + 81 ...... infinite terms - The sum will be infinite since you keep adding larger and larger terms.
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Karishma
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Re: For every integer k from 1 to 10, inclusive, the kth term of   [#permalink] 12 Oct 2015, 20:33

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