Find the missing number in the series 1, 4, 27, ?.

Introduction / Context:

This number series looks short but hides a powerful pattern based on exponents. Recognizing that the terms can be expressed as n raised to the power n provides the key insight. Such exponent based series often appear in competitive exams to test conceptual understanding of powers and sequences.

Given Data / Assumptions:

• The series is 1, 4, 27, ?.
• We must find a single missing term after 27.
• The numbers are 1, 4, 27, all of which are natural candidates for representation as powers.
• The pattern likely relates the base and exponent in a simple way.

Concept / Approach:

We test whether each term can be written as n^n for some integer n. If that works for the known terms, then we can continue with the next integer value to obtain the missing term. This method is a standard trick when an apparently irregular set of numbers can be interpreted as unusual powers such as 3^3, 4^4, and so on.

Step-by-Step Solution:

Step 1: Express 1 as a power: 1 = 1^1. Step 2: Express 4 as a power: 4 = 2^2. Step 3: Express 27 as a power: 27 = 3^3. Step 4: We see a clear pattern that the nth term is n^n, where n increases by 1 each step. Step 5: For the fourth term, n = 4, so the missing number must be 4^4 = 4 * 4 * 4 * 4 = 256.

Verification / Alternative check:

List the sequence using the formula n^n explicitly. For n = 1, we get 1^1 = 1. For n = 2, 2^2 = 4. For n = 3, 3^3 = 27. For n = 4, 4^4 = 256. Every term matches the given series up to the third term and provides a natural continuation for the fourth term. There is no need for a more complex pattern, since this simple exponent rule already explains all known values.

Why Other Options Are Wrong:

Option 125 is 5^3, which would not follow directly after 3^3 if we treat the exponent as tied to the base. Option 625 is 5^4, again not corresponding to the next integer n = 4 in the n^n pattern. Option 404 has no meaningful representation as n^n with small integers. Option 81 equals 3^4 or 9^2, but that does not extend the observed 1^1, 2^2, 3^3 structure. Therefore, 256 is the only option that aligns perfectly with the pattern n^n.

Common Pitfalls:

Some learners may first think of perfect squares or cubes separately without noticing that the base and exponent are the same for each term. Others may attempt to use differences or ratios, which in this case are irregular and unhelpful. A good habit when dealing with 1, 4, 27, and similar numbers is to test whether they are small integer powers like 1^1, 2^2, 3^3, 4^4, which quickly reveals the intended rule.

Final Answer:

The missing term that continues the pattern n^n is 256.