Complete the pair-doubling series: 4 → 8, 16 → 32, 64 → ( … ) Find the missing term that continues the same rule.

Introduction / Context:
This number-series question checks recognition of a simple deterministic rule applied consistently to each pair. Spotting the operation between numbers inside each pair allows us to extend the pattern and fill the blank correctly.

Given Data / Assumptions:

• Three ordered pairs are shown: (4, 8), (16, 32), (64, ?).
• Within each pair, the second number is derived from the first by the same rule.
• All numbers are integers and the sequence is monotonic within each pair.

Concept / Approach:
Look at the mapping inside each pair. If the second element is consistently a multiple of the first, the multiplier should be the same across pairs. Confirm that multiplier by testing the first two pairs, then apply it to the third pair to get the missing value.

Step-by-Step Solution:
Check pair 1: 4 → 8. This is 4 * 2 = 8.Check pair 2: 16 → 32. This is 16 * 2 = 32.Therefore, the rule inside each pair is “multiply by 2.”Apply the rule to pair 3: 64 → 64 * 2 = 128.

Verification / Alternative check:
The constant multiplier 2 preserves exact proportional growth. No other consistent single-step operation (addition, subtraction, division) fits both earlier pairs. Hence doubling is uniquely validated here.

Why Other Options Are Wrong:

• -128 / -192: introduce a sign change not used in earlier pairs.
• 192: would require multiplying 64 by 3, which contradicts the confirmed factor of 2.

Common Pitfalls:

• Mistaking a pair-wise rule for a cross-pair progression. Here each pair is independent but follows the same mapping.

Final Answer:
128