Consider the series 2, 16, 3, 81, 4, ?. Each pair of numbers appears to follow a mathematical rule. What number should replace the question mark so that the pattern is preserved?

Introduction / Context:
This question presents a short alternating series that consists of a small number followed by a larger one, repeated. Problems of this type are usually based on exponentiation, squares, cubes or simple functional relationships between each small number and the large number that follows it. The goal is to identify that relationship and then apply the same idea to the final pair involving the number 4 and the missing term.

Given Data / Assumptions:

• The series is: 2, 16, 3, 81, 4, ?
• Numbers appear in pairs: (2, 16), (3, 81), (4, ?).
• We assume the same rule applies to each pair.
• The rule is likely to involve powers of the first number in each pair.

Concept / Approach:
When we see a small integer followed by a much larger one, powers are a natural candidate: n^2, n^3, n^4 etc. Checking 2 and 16, and 3 and 81 quickly suggests that these are perfect fourth powers, since 2^4 = 16 and 3^4 = 81. If this pattern holds, then the next term should be 4 raised to the same power. This approach is straightforward and matches common exam patterns.

Step-by-Step Solution:
Step 1: Examine the first pair (2, 16). Compute 2^2 = 4, 2^3 = 8 and 2^4 = 16. The second element 16 equals 2^4. Step 2: Examine the second pair (3, 81). Compute 3^2 = 9, 3^3 = 27 and 3^4 = 81. The second element 81 equals 3^4. Step 3: Recognize the pattern: each larger term is the fourth power of the smaller term that precedes it. Step 4: Apply this pattern to the final pair (4, ?). Compute 4^4 = 4 * 4 * 4 * 4 = 16 * 16 = 256. Step 5: Therefore, the missing term must be 256.

Verification / Alternative check:
We can confirm that no other simple exponent pattern fits as neatly. If we tried squares, 2^2 = 4 and 3^2 = 9 would not match 16 and 81. Cubes 2^3 = 8, 3^3 = 27 also fail. The fourth power pattern exactly explains both known pairs, making it the most natural and reliable rule. Applying it to 4 gives a clean integer result, 256, strengthening our confidence.

Why Other Options Are Wrong:
Values like 99, 125 or 396 do not equal 4^k for any small integer k. In particular, 125 is 5^3, which would require the base in the pair to be 5, not 4. Similarly, 396 has no direct simple power relationship with 4. Because the existing pairs both follow a perfect fourth power pattern, any choice that does not equal 4^4 breaks the consistency of the sequence.

Common Pitfalls:
Some learners may look for differences or ratios across the entire line of numbers (for example, from 2 to 16 to 3 to 81) instead of respecting the natural pairing structure. Others may guess square or cube patterns without checking both given pairs. It is important to verify that the rule works for all existing examples before applying it to the missing term.

Final Answer:
Following the pattern that each second number is the fourth power of the first, the missing term corresponding to 4 is 256.