Question

the first four terms of a geometric sequence are
108, 36, 12, 4, ...
what is the common ratio?

-72
-3
3
\\(\frac{1}{3}\\)

Explanation:

Identify the terms of the geometric sequence

The first four terms of the geometric sequence are:

$$ a_1 = 108,\quad a_2 = 36,\quad a_3 = 12,\quad a_4 = 4 $$

Calculate the common ratio

The common ratio \(r\) is found by dividing any term by its preceding term:

$$ r = \frac{a_2}{a_1} = \frac{36}{108} = \frac{1}{3} $$

Verify with subsequent terms

$$ \frac{a_3}{a_2} = \frac{12}{36} = \frac{1}{3},\quad \frac{a_4}{a_3} = \frac{4}{12} = \frac{1}{3} $$

Answer:

• (A) \(-72\)
• (B) \(-3\)
• (C) \(3\)
• (D) \(\frac{1}{3}\) (Correct answer)