mushyyy wrote:
Given a sequence: \(a_1, \ a_2, \ a_3, \ ... \ a_{14}, \ a_{15}\)
In the sequence shown, \(a_n = a_{n-1}+k\), where \(2\leq{n}\leq{15}\) and \(k\) is a nonzero constant. How many of the terms in the sequence are greater than 10?
(1) \(a_1= 24\)
(2) \(a_8= 10\)
I love this question!!
Given: In the sequence shown, \(a_n = a_{n-1}+k\), where \(2\leq{n}\leq{15}\) and \(k\) is a nonzero constant. IMPORTANT: This sequence has exactly
15 terms.
Also, since k doesn't equal 0,
the sequence EITHER increases with each subsequent term OR decreases with each subsequent term.
Target question: How many of the terms in the sequence are greater than 10? Statement 1: \(a_1= 24\) This doesn't help us answer the target question.
Consider these two possible cases:
Case a: k = 2, which means each term is 2 greater than the previous term. So our sequence looks like this: 24, 26, 28, 30, etc, In this case, the answer to the target question is
all 15 terms are greater than 10Case b: k = -10, which means each term is 10 less than the previous term. So our sequence looks like this: 24, 14, 4, -6, -16, etc, In this case, the answer to the target question is
2 terms are greater than 10Since we cannot answer the
target question with certainty, statement 1 is NOT SUFFICIENT
Statement 2: \(a_8= 10\)IMPORTANT: Notice that \(a_8\) is the
middle term among all 15 terms.
We already know that
the sequence EITHER increases with each subsequent term OR decreases with each subsequent term. So let's consider each possible case:
Case a: The sequence INCREASES with each subsequent term. In this case, terms \(a_1\) to \(a_7\) will be less than 10, and terms \(a_9\) to \(a_{15}\) will be greater than 10. So, the answer to the target question is
7 terms are greater than 10Case b: The sequence DECREASES with each subsequent term. In this case, terms \(a_1\) to \(a_7\) will be greater than 10, and terms \(a_9\) to \(a_{15}\) will be less than 10. So, the answer to the target question is
7 terms are greater than 10Since both possible cases yield the SAME answer to the target question (
7 terms are greater than 10), we can answer the target question with certainty.
Statement 2 is SUFFICIENT
Answer: B
Cheers,
Brent