Quantative Review 2nd Edition, Problem Solving #133

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Can someone please explain to me how this problem works? Why do they re-write the last terms as (-1)(x-2)?

-Rick

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by selango » Sat Jul 31, 2010 7:11 pm
RickH,

Can you post the entire problem?
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by RickH » Sun Aug 01, 2010 7:43 am
I actually posted the wrong number it should be #107.

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

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by selango » Sun Aug 01, 2010 8:01 am
Sn=1/n - 1/n+1

S1=1-1/2=1/2

S2=1/2-1/3=1/6

S3=1/3-1/4=1/12

k=2,S1+S2=2/3

k=3,S1+S2+S3=3/4

So Sk=1-1/(k+1)

Is Sk>9/10

1-1/(k+1)>9/10

1/10>1/(k+1)

k+1>10

k>9?

stmt1,

k>10

So surely k>9

Suff

stmt2,

k<19

k can be k<9 or k>9

Not suff

Pick A
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by RickH » Mon Aug 02, 2010 11:42 am
can you please explain how you got this? I have been trying out but I don't see how.

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by anirban_lax » Mon Aug 02, 2010 2:23 pm
To understand how the sequence can be simplified you can just write out the various terms of the sequence putting in various values for n.
s1 = 1/1 - 1/2
s2 = 1/2 - 1/3
s3 = 1/3 - 1/4
s4 = 1/4 - 1/5
...
...
s(k-1)th term = 1/(k-1) - 1/k
sk = 1/k - 1/(k+1)

Now, if we add them up then we can see that values cancel out; -1/2 and +1/2, -1/3 and +1/3 and so on. The only terms that remain are 1/1 and -1/(k+1).
Therefore,
the problem boils down to deciding whether 1- 1/(k+1) is greater than 9/10 or not.
So, the inequality that we should be trying to solve is

1 - 1/(k+1) > 9/10

Subtract 1 from both sides,

-1/(k+1) > -1/10
Multiplying by -1, reverses the sign,

1/(k+1) < 1/10
Now cross multiply; this won't change the sign since k is positive and so is k+1.
10 < k+1
or, 9<k

So, the sum of the sequence will be greater than 9/10 if the value of k is greater than 9.

Statement 1 tells us that k>10. So, k ios definitely greater than 9 which means it guarantees that the sum of the sequence will be greater than 9/10.

Statement 2 tells us that k < 19. Now as long as k > 9 the sum of the sequence will be greater than 9/10 but when the value of k becomes less than 9, the sum will not be greater than 9/10. Hence, with statement 2 we have both possibilities. Thus this statement is not sufficient.

Hope, this helps you understand.

This question got me a little confused. Isn't 'sn' a standard notation supposed to mean the Sum of a sequence upto its nth term? To denote a sequence a simple 's' is used. Now, these are universal notations and GMAAC shouldn't be messing around with such standard notations, unless they explicitly mention something in the line of "there is a sequence that can be expressed as".

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Anirban

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by deeyah » Wed Aug 04, 2010 4:52 pm
Thanks for the explanation.