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.