Question

what is the common ratio of the sequence below?
\\(\frac{2}{3}, \frac{1}{6}, \frac{1}{24}, \frac{1}{96}, dots\\)
\\(\circ \frac{1}{9}\\)
\\(\circ \frac{1}{4}\\)
\\(\circ 4\\)
\\(\circ 9\\)

Explanation:

Step1: Recall common ratio formula

The common ratio \( r \) of a geometric sequence is found by \( r=\frac{a_{n + 1}}{a_n} \), where \( a_{n+1} \) is the next term and \( a_n \) is the current term.

Step2: Calculate ratio between first and second term

Take \( a_1=\frac{2}{3} \) and \( a_2 = \frac{1}{6} \). Then \( r=\frac{a_2}{a_1}=\frac{\frac{1}{6}}{\frac{2}{3}} \).
Simplify the division of fractions: \( \frac{1}{6}\div\frac{2}{3}=\frac{1}{6}\times\frac{3}{2}=\frac{3}{12}=\frac{1}{4} \).

Step3: Verify with other terms

Check \( a_3=\frac{1}{24} \) and \( a_2=\frac{1}{6} \). \( r=\frac{\frac{1}{24}}{\frac{1}{6}}=\frac{1}{24}\times6=\frac{6}{24}=\frac{1}{4} \).
Also check \( a_4=\frac{1}{96} \) and \( a_3=\frac{1}{24} \). \( r=\frac{\frac{1}{96}}{\frac{1}{24}}=\frac{1}{96}\times24=\frac{24}{96}=\frac{1}{4} \).

Answer:

\(\frac{1}{4}\) (corresponding to the option \(\frac{1}{4}\))