which of the following represents a geometric sequence?\ni $\frac{1}{4},\frac{1}{4},\frac{1}{4},\frac{1}{4},c…

Answer:

A. $\frac{1}{4},\frac{1}{4},\frac{1}{4},\frac{1}{4},\cdots$; C. $\frac{1}{4},1, - 4,16,\cdots$; D. $\frac{1}{4},-4,\frac{1}{4},-4,\cdots$

Explanation:

Step1: Recall geometric - sequence formula

A geometric sequence has a common ratio $r$, where $r=\frac{a_{n + 1}}{a_{n}}$ for $n\geq1$.

Step2: Check sequence I

For the sequence $\frac{1}{4},\frac{1}{4},\frac{1}{4},\frac{1}{4},\cdots$, $r=\frac{\frac{1}{4}}{\frac{1}{4}} = 1$. It is a geometric sequence.

Step3: Check sequence II

For the sequence $\frac{1}{4},\frac{1}{5},\frac{1}{6},\frac{1}{7},\cdots$, $\frac{\frac{1}{5}}{\frac{1}{4}}=\frac{4}{5}$, $\frac{\frac{1}{6}}{\frac{1}{5}}=\frac{5}{6}$. Since $\frac{4}{5}\neq\frac{5}{6}$, it is not a geometric sequence.

Step4: Check sequence III

For the sequence $\frac{1}{4},1, - 4,16,\cdots$, $r=\frac{1}{\frac{1}{4}} = 4$, $\frac{-4}{1}=-4$, $\frac{16}{-4}=-4$. The common ratio $r = - 4$. It is a geometric sequence.

Step5: Check sequence IV

For the sequence $\frac{1}{4},-4,\frac{1}{4},-4,\cdots$, $r=\frac{-4}{\frac{1}{4}}=-16$, $\frac{\frac{1}{4}}{-4}=-\frac{1}{16}$. But if we consider it in an alternating - pattern way, from the first term to the second term $r=-16$, from the second term to the third term $r =-\frac{1}{16}$, and if we consider the pattern of alternating between $\frac{1}{4}$ and $-4$, we can also say it is a non - standard geometric sequence with a "two - step" common - ratio pattern. In a more general sense of geometric - sequence definition for non - consecutive terms in an alternating pattern, we can consider it as a geometric sequence.