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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]  26 Apr 2012, 06:14
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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, 06:54, edited 1 time in total.
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Kudos [?]: 47250 [0], given: 7115

Re: Two sequence problems [#permalink]  26 Apr 2012, 06:54
Expert's post
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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Joined: 02 Sep 2009
Posts: 28754
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Kudos [?]: 47250 [2] , given: 7115

Re: The sequence s1, s2, s3,.....sn,...is such that Sn= [#permalink]  26 Apr 2012, 06:56
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Expert's post
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.

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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Kudos [?]: 47250 [0], given: 7115

Re: The sequence s1, s2, s3,.....sn,...is such that Sn= [#permalink]  26 Apr 2012, 06:57
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 $$n=10$$.

In case of any question please post it here: in-the-sequence-x0-x1-x2-xn-each-term-from-x1-to-xk-126564.html
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Re: The sequence s1, s2, s3,.....sn,...is such that Sn=   [#permalink] 26 Apr 2012, 06:57
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