Number series (alternating signs, doubling magnitude): 4, -8, 16, -32, 64, (…) Find the next term that preserves the alternating sign and geometric growth.

Introduction / Context:This is a classic alternating-sign geometric sequence. The absolute value doubles each step, while the sign flips positive/negative alternately.

Given Data / Assumptions:

• Terms: 4, -8, 16, -32, 64, (…)
• Common ratio in magnitude appears to be 2.
• Signs alternate +, −, +, −, +, …

Concept / Approach:A geometric pattern with consistent doubling implies next magnitude is previous magnitude * 2. The sign alternates each term, so follow the established sign order.

Step-by-Step Solution:

Verification / Alternative check:Express as term n: a1 = 4, a(n) = 4 * 2^(n−1) with sign (−1)^(n−1). For n = 6, a6 = −128, which matches.

Why Other Options Are Wrong:

• 128: Correct magnitude but wrong sign.
• 192, −192: Do not fit doubling; they imply ×3.

Common Pitfalls:Forgetting to alternate signs while applying the geometric factor. Track both magnitude and sign positions.

Final Answer:-128