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The sequence s1, s2, s3,.....sn,...is such that Sn= [#permalink]
26 Apr 2012, 06:14

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Question Stats:

100% (01:11) correct
0% (00:00) wrong based on 7 sessions

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_kis 3 greater than the previous term, and each term from x_k+1 to x_nis 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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Failing to plan is planning to fail.

Last edited by Bunuel on 26 Apr 2012, 06:54, edited 1 time in total.

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_kis 3 greater than the previous term, and each term from x_k+1 to x_nis 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!

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

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.

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.