1, 4, 9, 16, 25, ....

Answer: 36 is the next number in the sequence.

Method 1 (perfect squares): The terms are squares of consecutive integers: 1 = 1^2, 4 = 2^2, 9 = 3^2, 16 = 4^2, 25 = 5^2. Therefore the next term is 6^2 = 36.

Method 2 (differences): Look at differences between consecutive terms: 4 - 1 = 3, 9 - 4 = 5, 16 - 9 = 7, 25 - 16 = 9. These are consecutive odd numbers (3, 5, 7, 9), so the next difference is 11 and 25 + 11 = 36.

Conclusion: Both methods show the next number is 36.