To understand how the sequence works and determine the operation used, let's break down the pattern you've provided:

You have pairs of numbers, and there seem to be some operations (addition and subtraction) that modify these pairs. Let's start by identifying each transformation step by step.

The sequence provided is:

1. $(2, 3)$ → $-1 + 1$
2. $(1, 4)$ → $+1 + 3$
3. $(2, 7)$ → $+8 + 5$
4. $(10, 12)$ → $-9 - 9$

Analyzing each pair:

1. From $(2, 3)$ to $(1, 4)$:

   • First element: $2 - 1 = 1$
   • Second element: $3 + 1 = 4$

2. From $(1, 4)$ to $(2, 7)$:

   • First element: $1 + 1 = 2$
   • Second element: $4 + 3 = 7$

3. From $(2, 7)$ to $(10, 12)$:

   • First element: $2 + 8 = 10$
   • Second element: $7 + 5 = 12$

4. From $(10, 12)$ to ???:

   • The next pair operation would be:
     • First element: $10 - 9 = 1$
     • Second element: $12 - 9 = 3$

Conclusion:

The operation seems to alternate between adding and subtracting values to the pair of numbers. Specifically:

• For each pair, the first element is modified by adding or subtracting a number.
• The second element follows a similar pattern, but it increases by the sum of two values.

So, to predict the next pair after each transformation:

• For the first number: Apply either addition or subtraction of a given value.
• For the second number: Add or subtract a corresponding value to the second element.

Would you like more details on how to generalize this rule or explore more sequences like this?

Follow-up questions:

1. How would you extend this pattern for the next few pairs?
2. What would happen if you changed the sign of the operations?
3. Can you spot any connection between the values being added or subtracted?
4. Could you express the pattern mathematically with a formula?
5. How would you find the next sequence step without explicitly seeing the numbers?

Tip: Identifying alternating patterns often reveals hidden structures in number sequences!