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The sequence s1, s2, s3,.....sn,...is such that Sn=

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The sequence s1, s2, s3,.....sn,...is such that Sn= [#permalink]

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I don't know whether these problems have already been posted on the site, since I couldn't find the answers I will post them.

1) The sequence s_1, s_2, s_3, ..., s_n, ... is such that s_n=1/n - 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 [fraction]{9}{10}[/fraction]?
(1) k > 10
(2) k < 19


2) In the sequence \(x_0, x_1, x_2, ..., x_n,\) each term from \(x_1 to x_k\)is 3 greater than the previous term, and each term from \(x_k+1 to x_n\)is less than the previous term, where n and k are positive integers and k< n. If \(x_0 = x_n = 0\) and if \(x_k = 15\), what is the value of n?
A) 5
B) 6
C) 9
D) 10
E) 15

Please elaborate these problems as simple as possible! Thank you! :)
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Last edited by Bunuel on 26 Apr 2012, 07:54, edited 1 time in total.
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Re: Two sequence problems [#permalink]

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New post 26 Apr 2012, 07:54
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Stiv wrote:
I don't know whether these problems have already been posted on the site, since I couldn't find the answers I will post them.

1) The sequence \(s_1, s_2, s_3, ..., s_n, .., is such that [m]s_n=1/n - 1/n+1\) 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 [fraction]{9}{10}[/fraction]?
(1) k > 10
(2) k < 19


2) In the sequence \(x_0, x_1, x_2, ..., x_n,\) each term from \(x_1 to x_k\)is 3 greater than the previous term, and each term from \(x_k+1 to x_n\)is less than the previous term, where n and k are positive integers and k< n. If \(x_0 = x_n = 0\) and if \(x_k = 15\), what is the value of n?
A) 5
B) 6
C) 9
D) 10
E) 15

Please elaborate these problems as simple as possible! Thank you! :)


Two things:
1. Please post one question per topic;
2. Please post PS questions in the PS subforum: gmat-problem-solving-ps-140/ and DS questions in the DS subforum: gmat-data-sufficiency-ds-141/
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Re: The sequence s1, s2, s3,.....sn,...is such that Sn= [#permalink]

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New post 26 Apr 2012, 07:56
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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?

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.

In case of any question please post it here: the-sequence-s1-s2-s3-sn-is-such-that-sn-1-n-103947.html
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PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.


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Expert Post
Math Expert
User avatar
Joined: 02 Sep 2009
Posts: 34077
Followers: 6090

Kudos [?]: 76595 [0], given: 9976

Re: The sequence s1, s2, s3,.....sn,...is such that Sn= [#permalink]

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New post 26 Apr 2012, 07:57
Expert's post
In the sequence \(x_0, \ x_1, \ x_2, \ ... \ x_n\), each term from \(x_1\) to \(x_k\) is 3 greater than the previous term, and each term from \(x_{k+1}\) to \(x_n\) is 3 less than the previous term, where \(n\) and \(k\) are positive integers and \(k<n\). If \(x_0=x_n=0\) and if \(x_k=15\), what is the value of \(n\)?

A.5
B. 6
C. 9
D. 10
E. 15

Probably the easiest way will be to write down all the terms in the sequence from \(x_0=0\) to \(x_n=0\). Note that each term from from \(x_0=0\) to \(x_k=15\) is 3 greater than the previous and each term from \(x_{k+1}\) to \(x_n\) is 3 less than the previous term:

So we'll have: \(x_0=0\), 3, 6, 9, 12, \(x_k=15\), 12, 9, 6, 3, \(x_n=0\). So we have 11 terms from \(x_0\) to \(x_n\) thus \(n=10\).

Answer: D.

In case of any question please post it here: in-the-sequence-x0-x1-x2-xn-each-term-from-x1-to-xk-126564.html
_________________

New to the Math Forum?
Please read this: All You Need for Quant | PLEASE READ AND FOLLOW: 12 Rules for Posting!!!

Resources:
GMAT Math Book | Triangles | Polygons | Coordinate Geometry | Factorials | Circles | Number Theory | Remainders; 8. Overlapping Sets | PDF of Math Book; 10. Remainders | GMAT Prep Software Analysis | SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS) | Tricky questions from previous years.

Collection of Questions:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.


What are GMAT Club Tests?
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Re: The sequence s1, s2, s3,.....sn,...is such that Sn=   [#permalink] 26 Apr 2012, 07:57
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