In a sequence S, each term, after the first term, is obtained by

Assume that the first term in the sequence is x

Hence, the sequence can be represented as

\(x, \quad xk, \quad xk^2, \quad x^3, \quad ...... \quad x^{n-1}\)

\(n^{\text{th}} \text{term of the sequence}= x^{n-1}\)

Statement 1

(1) The 8th term in the sequence is 32 and the 2nd term in the sequence is \(\frac{1}{2}\)


8th term = \(xk^7\) = 32
2nd term = \(xk = \frac{1}{2}\)

\(\frac{\text{8th term} }{ \text{2nd term}} = k^6 = 64\)

\(k = \pm 2\)

k = +2; x = 1/4

k = -2; x = -1/4

As we have two combinations, we cannot predict many terms in the sequence are less than 1.

Statement 2

\(xk^4 = 4\)

Clearly insufficient.

Eliminate B

Combined

As \(xk^4 = 4\)

\(k^4\) is positive, therefore for the multiplication to be a positive term x has to be positive.

Therefore x = 1/4 & k = 2

Once, we have the values in place, we can formulate the sequence and count the number of terms that are less than 1.

Sufficient.

Option C