What comes next? 1, 4, 4, 16, 64, ?

The next number in the sequence is 256. To understand how we arrive at this answer, it is essential to analyze the pattern exhibited by the given numbers: 1, 4, 4, 16, 64.

Starting with the sequence, we can observe that the first two numbers, 1 and 4, can be related to the powers of 2. Specifically, 1 can be expressed as \(2^0\) and 4 as \(2^2\). The next number is also 4, which can be seen as \(2^2\) again. Following this, 16 can be recognized as \(2^4\) and 64 as \(2^6\). Notably, the exponents are increasing in an arithmetic pattern: 0, 2, 2, 4, 6.

To derive the next term, we observe that the exponents are alternating between increasing by 2 and remaining the same. After reaching 6, the next logical exponent, following the established pattern, should be 8. Thus, the next number in the sequence is \(2^8\), which equals 256. This systematic approach not only reveals the next term but also reinforces the importance of recognizing patterns and relationships in sequences, a skill that is crucial for success in competitive exams.

In summary, when determining what comes next in this sequence, recognizing the underlying mathematical relationships and patterns is vital. By applying these principles, we can confidently conclude that the next number is 256.