A sequence of terms a1, a2 ,a3, ..., a(m-1), am, is given by

A sequence of non-zero terms \(a_1\), \(a_2\), \(a_3\), ..., \(a_{m-1}\), \(a_m\), is given by \(a_k=(a_{k-1})^2(a_{k-2})\) for every k>2. If m=12, then how many terms in the given sequence are positive?

From above:
\(a_3=(a_2)^2*a_1\);
\(a_4=(a_3)^2*a_2\);
...

(1) \(a_3\) is positive --> \(a_3=(a_2)^2*a_1=positive\) --> \(a_1=positive\). Now, if \(a_1=a_2=1\), then ALL 12 terms in the sequence will be positive but if \(a_1=1\), and \(a_2=-1\) (\(a_3=(a_2)^2*a_1=(-1)^2*1=1=positive\)), then not all the terms in the sequence will be positive. Not sufficient.

(2) \(a_4\) is positive --> \(a_4=(a_3)^2*a_2=positive\) --> \(a_2=positive\). The same here: if \(a_1=a_2=1\), then ALL 12 terms in the sequence will be positive but if \(a_1=-1\), and \(a_2=1\) (\(a_3=(a_2)^2*a_1=(1)^2*(-1)=-1\) and \(a_4=(a_3)^2*a_2=(-1)^2*1=1=positive\)), then not all the terms in the sequence will be positive. Not sufficient.

(1)+(2) From above we have that \(a_1=positive\) and \(a_2=positive\). Therefore, all 12 terms of the sequence are positive. Sufficient.

Answer: C.

Hope it's clear.
_________________

Then.. The information for "every k>2" is irrelevant right

Posted from my mobile device

"A sequence ... is given by \(a_k=(a_{k-1})^2(a_{k-2})\) for every k>2" means that the given formula applies for the terms starting \(a_3\).
_________________

Oh I get it, thanks!

Although the Answer is correct..but as I see the question Posted and the question in the image are different. Considering the question in the image a1 = +ve, a2=-ve, a3=+ve, a4=-ve and so on...Therefore, there will be 6 +ve terms in the sequence...

How can we say that because a1 and a2 are positive all terms will be positive?

\(a_3=(a_2)^2*a_1\);
\(a_4=(a_3)^2*a_2\);
...

Now, if a1 and a2 are both positive can a3 be negative? a4? an?
_________________

Statement 2 in the question and in the screenshot are different! is \(a_4\) positive or negative?

The discussion is on the question which says that \(a_4\) is positive.
_________________

Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
_________________