Question

what is the common ratio of the geometric sequence below?

\\(-2, 4, -8, 16, -32, \dots\\)

• \\(-\frac{4}{2}\\)
• \\(-\frac{2}{4}\\)
• \\(\frac{2}{4}\\)
• \\(\frac{4}{2}\\)

Explanation:

Identify consecutive terms of the sequence

The given geometric sequence is:

$$ a_1 = -2,\quad a_2 = 4,\quad a_3 = -8,\quad a_4 = 16,\quad a_5 = -32 $$

Calculate the common ratio

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

$$ r = \frac{a_2}{a_1} = \frac{4}{-2} = -\frac{4}{2} $$

Verify with subsequent terms

$$ r = \frac{a_3}{a_2} = \frac{-8}{4} = -2 $$

The calculated ratio matches the option written as \(-\frac{4}{2}\).

Answer:

• (A) -\frac{4}{2} (Correct answer)
• (B) -\frac{2}{4}
• (C) \frac{2}{4}
• (D) \frac{4}{2}