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The sequence s1, s2, s3, ..., sn, ... is such that Sn= 1/n - 1/(n + 1)

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The sequence s1, s2, s3, ..., sn, ... is such that Sn= 1/n - 1/(n + 1)  [#permalink]

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New post 17 Aug 2010, 12:46
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The sequence s1, s2, s3, ..., sn, ... is such that \(S_n= \frac{1}{n} - \frac{1}{n+1}\) for all integers \(n\geq 1\). If k is a positive integer, is the sum of the first k terms of the sequence greater than 9/10?

(1) k > 10
(2) k < 19

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Re: The sequence s1, s2, s3, ..., sn, ... is such that Sn= 1/n - 1/(n + 1)  [#permalink]

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New post 17 Aug 2010, 13:02
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6
(1) Take K as 11.
So, Sum = S1 + S2 + S3 + S4 + S5+ S6 + S7 + S8 + S9 + S10 + S11
Where,
S1 = 1 - (1/2)
S2 = (1/2) - (1/3)
S3 = (1/3) - (1/4) ...
S11 = (1/11) - (1/12)

Which implies, Sum = 1 - (1/12) /* The terms like +1/2, -1/2, +1/3, -1/3 will be added to zero. Only the first and last numbers remains */
==> Sum = 1 - 0.0XXXX > 9/10

If you take k as 12, the SUM = 1 - (1/13) which is again > 0.9 Hence SUFFICIENT

(2) K < 19
Consider K as 2. Then the sum is = 1 - (1/2) + (1/2) - (1/3) = 1 - (1/3) which is Less than 9/10
Consider K as 11, Then the sum is greater than 9/10 /* We already proved this in (1) above */
Hence (2) is In Sufficient
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Re: The sequence s1, s2, s3, ..., sn, ... is such that Sn= 1/n - 1/(n + 1)  [#permalink]

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New post 17 Aug 2010, 13:22
If you notice, the sum is simply given by Sn = 1 -1/(n+1) = n/(n+1). hence S9 = 9/10. Adding further terms only increases the sum - hence S10 > S9 etc...
1) k>10, clearly sufficient as S10>9/10 = S9
2) k<19, cant say much K can be 1 = 1/2<9/10 but K=11 makes 11/12>9/10
Hence A
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Re: The sequence s1, s2, s3, ..., sn, ... is such that Sn= 1/n - 1/(n + 1)  [#permalink]

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New post 17 Aug 2010, 13:35
2
I am not a master in choosing the right number, but as i read & practising the same ... Choose the numbers which are closer to the lowest range & Highest range first.

Example:
(1) K > 10.
The lowest range number here is: 11. If 11 is sufficient then take 12. If 12 is also sufficient find out if you can make a generalized statement as Sufficient?

The highest range number is: Infinity

(2) K < 19
The lowest range number (according to problem specificaitons) is: 1
The highest range number is: 18

Cheers!
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Re: The sequence s1, s2, s3, ..., sn, ... is such that Sn= 1/n - 1/(n + 1)  [#permalink]

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New post 29 Oct 2010, 21:09
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satishreddy wrote:
The sequence s1, s2, s3,.....sn,...is such that Sn= (1/n) - (1/(n+1)) for all integers n>=1. If k is a positive integer, is the sum of the first k terms of the sequence greater than 9/10?
1) k > 10
2) k < 19


Given: \(s_n=\frac{1}{n}-\frac{1}{n+1}\) for \(n\geq{1}\). So:
\(s_1=1-\frac{1}{2}\);
\(s_2=\frac{1}{2}-\frac{1}{3}\);
\(s_3=\frac{1}{3}-\frac{1}{4}\);
...

If you sum the above 3 terms you'll get: \(s_1+s_2+s_3=(1-\frac{1}{2})+(\frac{1}{2}-\frac{1}{3})+(\frac{1}{3}-\frac{1}{4})=1-\frac{1}{4}\) (everything but the first and the last numbers will cancel out). So the sum of first \(k\) terms is fgiven by the formula \(sum_k=1-\frac{1}{k+1}\).

Question: is \(sum_k=1-\frac{1}{k+1}>\frac{9}{10}\)? --> is \(\frac{k}{k+1}>\frac{9}{10}\)? --> is \(k>9\)?

(1) k > 10. Sufficient.
(2) k < 19. Not sufficient.

Answer: A.
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The sequence s1, s2, s3, ..., sn, ... is such that Sn= 1/n - 1/(n + 1)  [#permalink]

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Re: The sequence s1, s2, s3, ..., sn, ... is such that Sn= 1/n - 1/(n + 1)  [#permalink]

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New post 17 Feb 2011, 10:23
Good Solution Bunuel..
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Re: The sequence s1, s2, s3, ..., sn, ... is such that Sn= 1/n - 1/(n + 1)  [#permalink]

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New post 03 Mar 2011, 10:24
1
S1=1/1-1/2= 1-1/2
so, Sumk=1-1/k+1
---->k/k+1>9/10
---->k>10
A sufficient
B not sufficient
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Re: The sequence s1, s2, s3, ..., sn, ... is such that Sn= 1/n - 1/(n + 1)  [#permalink]

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New post 04 Mar 2011, 12:09
satishreddy wrote:
The sequence s1, s2, s3,.....sn,...is such that Sn= (1/n) - (1/(n+1)) for all integers n>=1. If k is a positive integer, is the sum of the first k terms of the sequence greater than 9/10?
1) k > 10
2) k < 19


I) (1/n) - (1/n+1) where N>=1; k = sum of the sequence;

n=1; 1/1 - (1/1+1) = 1-(1/2) = 1/2
n=2; 1/2 - (1/2+1) = 1/2-1/3
n=3; 1/3 - (1/3+1) = 1/3-1/4

Sum of first 3 = 1/2 + (1/2-1/3) + (1/3 - 1/4) = 1 - 1/4 = First term - last term for sequence

Sum of N = 1 - (1/k+1) > 9/10
(k+1)/(k+1)-(1/k+1) > 9/10
k/(k+1) > 9/10
10k>9k+9?
k>9?

I)sufficient
II) sometimes yes sometimes no; insufficient

"A"
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Re: The sequence s1, s2, s3, ..., sn, ... is such that Sn= 1/n - 1/(n + 1)  [#permalink]

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New post 05 Mar 2011, 16:50
Good Question whoever posted it.

Great solution by Bunuel. Good that you didn't solve the sn formula initially to 1/n(n+1) . that way its easy to cancel out terms.


Bunuel wrote:
satishreddy wrote:
The sequence s1, s2, s3,.....sn,...is such that Sn= (1/n) - (1/(n+1)) for all integers n>=1. If k is a positive integer, is the sum of the first k terms of the sequence greater than 9/10?
1) k > 10
2) k < 19


Given: \(s_n=\frac{1}{n}-\frac{1}{n+1}\) for \(n\geq{1}\). So:
\(s_1=1-\frac{1}{2}\);
\(s_2=\frac{1}{2}-\frac{1}{3}\);
\(s_3=\frac{1}{3}-\frac{1}{4}\);
...

If you sum the above 3 terms you'll get: \(s_1+s_2+s_3=(1-\frac{1}{2})+(\frac{1}{2}-\frac{1}{3})+(\frac{1}{3}-\frac{1}{4})=1-\frac{1}{4}\) (everything but the first and the last numbers will cancel out). So the sum of first \(k\) terms is fgiven by the formula \(sum_k=1-\frac{1}{k+1}\).

Question: is \(sum_k=1-\frac{1}{k+1}>\frac{9}{10}\)? --> is \(\frac{k}{k+1}>\frac{9}{10}\)? --> is \(k>9\)?

(1) k > 10. Sufficient.
(2) k < 19. Not sufficient.

Answer: A.
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Re: The sequence s1, s2, s3, ..., sn, ... is such that Sn= 1/n - 1/(n + 1)  [#permalink]

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New post 01 Jan 2012, 08:12
mainhoon wrote:
If you notice, the sum is simply given by Sn = 1 -1/(n+1) = n/(n+1). hence S9 = 9/10. Adding further terms only increases the sum - hence S10 > S9 etc...
1) k>10, clearly sufficient as S10>9/10 = S9
2) k<19, cant say much K can be 1 = 1/2<9/10 but K=11 makes 11/12>9/10
Hence A

I don't think sn=1/n-1/(n+1) is equal to n/(n+1), it's equal to 1/n(n+1).

I find that sequence problems come down to pattern recognition, nravi's solution is probably what it's supposed to look like, you have to see that the terms other than 1 and 1/12 cancel each other out.
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Re: The sequence s1, s2, s3, ..., sn, ... is such that Sn= 1/n - 1/(n + 1)  [#permalink]

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New post 01 Jan 2012, 09:47
1
BN1989 wrote:
mainhoon wrote:
If you notice, the sum is simply given by Sn = 1 -1/(n+1) = n/(n+1). hence S9 = 9/10. Adding further terms only increases the sum - hence S10 > S9 etc...
1) k>10, clearly sufficient as S10>9/10 = S9
2) k<19, cant say much K can be 1 = 1/2<9/10 but K=11 makes 11/12>9/10
Hence A

I don't think sn=1/n-1/(n+1) is equal to n/(n+1), it's equal to 1/n(n+1).

I find that sequence problems come down to pattern recognition, nravi's solution is probably what it's supposed to look like, you have to see that the terms other than 1 and 1/12 cancel each other out.


sum of \([\frac{1}{n} - \frac{1}{(n+1)}]\) from = 1 to n=n => \(\frac{n}{(n+1)}\)
\(1-\frac{1}{2} + \frac{1}{2}-\frac{1}{3} + \frac{1}{3}-\frac{1}{4} + ............... \frac{1}{n} -\frac{1}{(n+1)}\) = \(1 - \frac{1}{(n+1)}\) =\(\frac{n}{(n+1)}\)
because all the terms between 1st and last are cancelled.
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Re: The sequence s1, s2, s3, ..., sn, ... is such that Sn= 1/n - 1/(n + 1)  [#permalink]

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New post 07 Jan 2012, 16:08
Tricky one, needs so serious rephrasing of the stem:

Rephrase: for the k number of elements, the sum of all the terms will be = 1/x - 1/(x+k) (try yourself! - intermediate terms cancel out)

1. Say k = 11, we get 1/1 - 1/(1+11) = 11/12 > 9/10. Suff
2. k<19. it could be k=11 as above or k=8, where you get 1/1 - 1/(1+8) = 8/9 < 9/10. Insuff.

A.
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Re: The sequence s1, s2, s3, ..., sn, ... is such that Sn= 1/n - 1/(n + 1)  [#permalink]

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New post 29 Mar 2012, 04:57
GMAT Club Legend - awesome work. Really appreciate the amount of effort you put in when laying out your answers... greatly helps the mere mortals!
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Re: The sequence s1, s2, s3, ..., sn, ... is such that Sn= 1/n - 1/(n + 1)  [#permalink]

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New post 22 Dec 2012, 19:44
bunuel,
Please let us know how to solve this one using the A.P formula.
Regards,
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Re: The sequence s1, s2, s3, ..., sn, ... is such that Sn= 1/n - 1/(n + 1)  [#permalink]

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Re: The sequence s1, s2, s3, ..., sn, ... is such that Sn= 1/n - 1/(n + 1)  [#permalink]

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New post 17 Oct 2016, 17:02
Bunuel wrote:
satishreddy wrote:
The sequence s1, s2, s3,.....sn,...is such that Sn= (1/n) - (1/(n+1)) for all integers n>=1. If k is a positive integer, is the sum of the first k terms of the sequence greater than 9/10?
1) k > 10
2) k < 19


Given: \(s_n=\frac{1}{n}-\frac{1}{n+1}\) for \(n\geq{1}\). So:
\(s_1=1-\frac{1}{2}\);
\(s_2=\frac{1}{2}-\frac{1}{3}\);
\(s_3=\frac{1}{3}-\frac{1}{4}\);
...

If you sum the above 3 terms you'll get: \(s_1+s_2+s_3=(1-\frac{1}{2})+(\frac{1}{2}-\frac{1}{3})+(\frac{1}{3}-\frac{1}{4})=1-\frac{1}{4}\) (everything but the first and the last numbers will cancel out). So the sum of first \(k\) terms is fgiven by the formula \(sum_k=1-\frac{1}{k+1}\).

Question: is \(sum_k=1-\frac{1}{k+1}>\frac{9}{10}\)? --> is \(\frac{k}{k+1}>\frac{9}{10}\)? --> is \(k>9\)?

(1) k > 10. Sufficient.
(2) k < 19. Not sufficient.

Answer: A.


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Re: The sequence s1, s2, s3, ..., sn, ... is such that Sn= 1/n - 1/(n + 1)  [#permalink]

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New post 18 Oct 2016, 02:00
oasis90 wrote:
Bunuel wrote:
satishreddy wrote:
The sequence s1, s2, s3,.....sn,...is such that Sn= (1/n) - (1/(n+1)) for all integers n>=1. If k is a positive integer, is the sum of the first k terms of the sequence greater than 9/10?
1) k > 10
2) k < 19


Given: \(s_n=\frac{1}{n}-\frac{1}{n+1}\) for \(n\geq{1}\). So:
\(s_1=1-\frac{1}{2}\);
\(s_2=\frac{1}{2}-\frac{1}{3}\);
\(s_3=\frac{1}{3}-\frac{1}{4}\);
...

If you sum the above 3 terms you'll get: \(s_1+s_2+s_3=(1-\frac{1}{2})+(\frac{1}{2}-\frac{1}{3})+(\frac{1}{3}-\frac{1}{4})=1-\frac{1}{4}\) (everything but the first and the last numbers will cancel out). So the sum of first \(k\) terms is fgiven by the formula \(sum_k=1-\frac{1}{k+1}\).

Question: is \(sum_k=1-\frac{1}{k+1}>\frac{9}{10}\)? --> is \(\frac{k}{k+1}>\frac{9}{10}\)? --> is \(k>9\)?

(1) k > 10. Sufficient.
(2) k < 19. Not sufficient.

Answer: A.


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Re: The sequence s1, s2, s3, ..., sn, ... is such that Sn= 1/n - 1/(n + 1)  [#permalink]

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New post 03 Jun 2017, 21:04
metallicafan wrote:
Please your help:

The sequence s1, s2, s3,.....sn,...is such that \(Sn= (1/n) - (1/(n+1))\) for all integers \(n>=1\). If k is a positive integer, is the sum of the first k terms of the sequence greater than \(9/10\)?

1) k > 10
2) k < 19


This problem is best tackled by identifying the first term.

In this case, S1 = 1/1 - 1/2.

S2 = 1/2 - 1/3

S3 = 1/3 - 1/4

If you add them together, all but the first and last terms cancel. So to determine if the first k terms are greater than 9/10, we need to determine if 1 - 1/(1+k) > 9/10.

In other words, is 1/(1+k) < 1/10?

10 < 1+k?

9<k?

Statement 1) K > 10. Sufficient.

Statement 2) K < 19. Not sufficient.
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Re: The sequence s1, s2, s3, ..., sn, ... is such that Sn= 1/n - 1/(n + 1)  [#permalink]

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New post 03 Jun 2017, 21:52
metallicafan wrote:
Please your help:

The sequence s1, s2, s3,.....sn,...is such that \(Sn= (1/n) - (1/(n+1))\) for all integers \(n>=1\). If k is a positive integer, is the sum of the first k terms of the sequence greater than \(9/10\)?

1) k > 10
2) k < 19


1) Lets assume the value of k to be 11

\(Sn= (1/n) - (1/(n+1))\) means \(Sn = (1-(1/(n+1))\), For 11 it will be 11/12 which is greater than 9/10. Any integer greater than 11 will also yield a fraction which will be greater than 9/10 hence sufficient

2) Lets assume the value of k to be 18
Then \(Sn = (1-(1/(n+1))\), For 18 it will be 18/19 which is greater than 9/10 but
if the value of K is 8 then Sn = 8/9 which is less than 9/10 hence insufficient

Answer will be A
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Re: The sequence s1, s2, s3, ..., sn, ... is such that Sn= 1/n - 1/(n + 1) &nbs [#permalink] 03 Jun 2017, 21:52

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