In the sequence d_1, d_2, d_3, . . ., for all positive integers n, d_

Given: c

Target question: Is \(d_k > 0\)?
This is a good candidate for rephrasing the target question.

Take: \(d_n=n^2 − 25n + 150\)
The factor to get: \(d_n=(n-10)(n-15)\)
Now recognize that:
-When \(15 < n\), we get: \(d_n=(n-10)(n-15)=(positive)(positive)= positive\)
-When \(10 < n < 15\), we get: \(d_n=(n-10)(n-15)=(positive)(negative)=negative\)
-When \(0<n < 10\), we get: \(d_n=(n-10)(n-15)=(negative)(negative)=positive\)

As we can see, \(d_k > 0\) when \(15 < k\) OR when \(k < 10\).
So....
REPHRASED target question: Is k EITHER greater than 15 OR less than 10?
Aside: the video below has tips on rephrasing the target question

When I scan the two statements, they both feel insufficient, AND I’m pretty sure I can identify some cases with conflicting answers to the REPHRASED target question. So, I’m going to head straight to……

Statements 1 and 2 combined
Statement 1 tells us that \(5 < k < 25\)
Statement 2 tells us that \(k > 10\)
Case a: If \(k = 20\), the answer to the REPHRASED target question is YES
Case b: If \(k = 12\), the answer to the REPHRASED target question is NO
Since we can’t answer the REPHRASED target question with certainty, the combined statements are NOT SUFFICIENT

Answer: E

VIDEO ON REPHRASING THE TARGET QUESTION:

The question has changed n to k as n could be anything 1, 2 etc, and now we are looking for: whether kth term >0?

So \(d_k\) = \(k^2\) − 25k + 150=\(k^2-15k-10k+150=(k-15)(k-10)\).

That is, the question becomes: Is (k-15)(k-10)>0?
When will this be true?
Case 1: When k<10, both k-10 and k-15 are negative so (k-10)(k-15) will be positive
Case 2: When 10<k<15, k-10>0 and k-15<0, so (k-10)(k-15) will be negative.
Case 3: When k>15, both k-10 and k-15 are positive so (k-10)(k-15) will be positive

Let us see whether we can find a definite range as per above cases of k.

1) 5 < k < 25
All 3 cases possible
Insufficient

2) k > 10
Again case 2 and case 3 are possible, giving us both yes and no as the answer
Insufficient


Combined
Range= 10<k<25.
Again cases 2 and 3 possible
10<k<15…..NO
15<k<25…..Yes
Insufficient



E