sequence

[mehrasa]

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

[spoiler]OA:A[/spoiler]
please tell me how we can simplify this sequence.. thnx

[cans]

S1 = 1/1..
s2 = 1/2 - 1/1
s3 = 1/3 - 1/2
.
.
sn = 1/n - 1/(n-1)
i think for n=1, s1 should be 1 and not 1/1 - 1/0 (as 1/0 is not defined)
thus sum for n integers: 1/n
b) k<19. if n=1, sum=1 > 9/10 and if k=10, sum=1/10 <9/10 Insufficient
a) k>10. k=11, 1/11 <9/10... as k increases, sum will decrease. Thus sufficient.
IMO A

[mehrasa]

I had a typo in the sequence dear cans.. the sequence is Sn= 1/n- 1/n+1

[cans]

Then its like:
S1 = 1/1 - 1/2
s2 = 1/2 - 1/3
s3 = 1/3 - 1/4
.
.
sn = 1/n - 1/(n+1)
thus sum of first n = 1 - 1/(n+1)
a) when k>10, k=11 means sum = 1 - 1/12 = 11/12 > 9/10
as k increases, 1/(n+1) decreases and thus sum increases.
Thus sufficient
b) k<19. at k=11, sum>9/10. but when n=1, sum=1/2 < 9/10. Insufficient

[GMATGuruNY]

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

Write out enough of the sequence to see the pattern:
S(1) = 1 - 1/2
S(2) = 1/2 - 1/3
S(3) = 1/3 - 1/4
etc.

Sum of the first 2 terms:
S(1) + S(2) = (1 - 1/2) + (1/2 - 1/3) = 1 - 1/3.

Sum of the first 3 terms:
S(1) + S(2) + S(3) = (1 - 1/2) + (1/2 - 1/3) + (1/3 - 1/4) = 1 - 1/4.

When the terms are added, every term between 1 and the last fraction cancels out.

In the sum of the first 2 terms, the last fraction = -1/3.
In the sum of the first 3 terms, the last fraction = -1/4.
Using this logic:
In the sum of the first k terms, the last fraction = -1/(k+1).

Thus, the sum of the first k terms = 1 - 1/(k+1).

Question rephrased: Is 1 - 1/(k+1) > 9/10?

Statement 1: k>10.
Let k=11.
Sum = 1 - 1/(11+1) = 11/12.
11/12 > 9/10.

Let k=12.
Sum = 1 - 1/(12+1) = 12/13.
12/13 > 9/10.

As k increases, so does the sum.
Since the smallest possible sum in statement 1 is 11/12, the sum will always be greater than 9/10.
Sufficient.

Statement 2: k<19.
Let k=1:
Sum = 1 - 1/2 = 1/2.
1/2 < 9/10.

Let k=11.
Sum = 1 - 1/(11+1) = 11/12.
11/12 > 9/10.

Since the sum can be less than 9/10 or greater than 9/10, insufficient.

The correct answer is A.

[navami]

IMO A