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by singhsa » Wed Sep 22, 2010 9:22 pm
a1,a2,a3...a15
In the sequence shown, a(n) = a(n-1)+k, where 2<=n<=15 and k is a nonzero constant. How many of the terms in the sequence are greater than 10?

(1) a1 = 24
(2) a8 = 10

OA-
B
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Source: — Data Sufficiency |

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by Rahul@gurome » Wed Sep 22, 2010 9:56 pm
a1,a2,a3...a15
In the sequence shown, a(n) = a(n-1)+k, where 2<=n<=15 and k is a nonzero constant. How many of the terms in the sequence are greater than 10?

(1) a1 = 24
(2) a8 = 10

Solution:
Note that k can be either positive or negative,
If k is positive, the sequence increases and if k is negative, sequence decreases.
Consider first (1) alone.
a1 = 24.
If k = 1, a2 = 25, a3 = 26,.......a15 = 38.
So all 15 terms are greater than 10.
Next let k = -10,
So a2 = 14, a3 = 4,.......
So only a1 and a2 or 2 terms are greater than 10.
Since nothing definite can be said (1) alone is not sufficient.
Next consider (2) alone.
a8 = 10.
Let k be positive.
Then a9= 10+k >10, a10 = 10+2k >10, .....a15 = 10 + 7k > 10.
So exactly 7 terms are > 10.
Next let k be negative.
So a7 = 10 - k > 10 because k is negative.
Similarly a6, a5, a4,.a1 are all greater than 10.
Again exactly 7 terms will be greater than 10.
So (2) alone is sufficient.

The correct answer is (B).
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by singhsa » Thu Sep 23, 2010 9:30 pm
Rahul@gurome wrote:a1,a2,a3...a15
In the sequence shown, a(n) = a(n-1)+k, where 2<=n<=15 and k is a nonzero constant. How many of the terms in the sequence are greater than 10?

(1) a1 = 24
(2) a8 = 10

Solution:
Note that k can be either positive or negative,
If k is positive, the sequence increases and if k is negative, sequence decreases.
Consider first (1) alone.
a1 = 24.
If k = 1, a2 = 25, a3 = 26,.......a15 = 38.
So all 15 terms are greater than 10.
Next let k = -10,
So a2 = 14, a3 = 4,.......
So only a1 and a2 or 2 terms are greater than 10.
Since nothing definite can be said (1) alone is not sufficient.
Next consider (2) alone.
a8 = 10.
Let k be positive.
Then a9= 10+k >10, a10 = 10+2k >10, .....a15 = 10 + 7k > 10.
So exactly 7 terms are > 10.
Next let k be negative.
So a7 = 10 - k > 10 because k is negative.
Similarly a6, a5, a4,.a1 are all greater than 10.
Again exactly 7 terms will be greater than 10.
So (2) alone is sufficient.

The correct answer is (B).
Hey Rahul. I have totally understood the way you've attacked this question but there's one thing that's bothering me.

Why are we assuming k>-10 in cases where we've considered k to be negative.

In the second case, if k=-20, all the terms would be negative. How can we assume a7=10-k to be positive. K can be anything. Please Explain.

Thanks.
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by Rahul@gurome » Thu Sep 23, 2010 10:35 pm
We are nowhere assuming that k > -10 where k is negative.
By statement (2) a8 = 10.
Let k be negative such that k = -b where b is positive.
So a7 = 10 - (-b) = 10+b = 10 +( a positive quantity). Or a7 > 10.
a6 = a7 - k = a7 + b = 10+b+b = 10+2b = 10 + (a positive quantity). Or a6>10.
You can similarly deduce that a5, a4, a3, a2, a1 are all greater than 10 if k is negative
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