how do you figure out the pattern of 1,4,9,16?

One approach is to analyze the differences between consecutive terms. Let's calculate the differences between each pair of consecutive terms:

4 - 1 = 3
9 - 4 = 5
16 - 9 = 7

By observing the differences, you can see that they are increasing by 2 each time. Starting with 3, then 5, and finally 7. This indicates that the original sequence may be the result of adding consecutive odd numbers.

To verify this, let's start with the first term, 1, and incrementally add consecutive odd numbers: 1, 1+3=4, 4+5=9, 9+7=16. As you can see, the pattern holds.

Thus, the pattern of the sequence 1, 4, 9, 16 is the result of squaring consecutive natural numbers: 1^2 = 1, 2^2 = 4, 3^2 = 9, 4^2 = 16.

By understanding how to analyze the differences and test relationships between consecutive terms, you can solve various types of pattern questions.