It looks like you're asking how a sequence of integer pairs transitions from one pair to the next and how the pair (+4, +1) was derived. Let's analyze the pattern:

Given pairs:

• (+5, -1)
• (+1, +4)
• (+2, -1)
• (-10, -10)

The next pair is (+4, +1). To find out how this pair is derived, we need to investigate if there is a clear pattern in the differences between consecutive terms of each pair:

First Integer of Each Pair:

• From +5 to +1: $1 - 5 = -4$
• From +1 to +2: $2 - 1 = +1$
• From +2 to -10: $-10 - 2 = -12$

The differences between the first integers are: -4, +1, and -12.

We don’t see an immediate linear pattern here, but the next first integer is +4, so let’s look at the second part to confirm.

Second Integer of Each Pair:

• From -1 to +4: $4 - (-1) = +5$
• From +4 to -1: $-1 - 4 = -5$
• From -1 to -10: $-10 - (-1) = -9$

The differences between the second integers are: +5, -5, and -9.

Again, there’s no simple arithmetic progression. However, if we apply similar large changes, they might have just chosen the next reasonable values.

So, the logic appears to be that the next first integer could increase or stabilize after the large jump of -12, leading to +4. Meanwhile, the second integer also stabilizes around smaller changes after the large jumps, resulting in +1.

Conclusion:

It seems the pattern involves alternating large and small changes, and the jump to (+4, +1) follows this more relaxed progression after the extreme changes in previous steps.

Would you like further details or have any questions? Here are 5 relative questions to expand on this:

1. How could we model this sequence with a different mathematical approach (e.g., using recursion)?
2. What are other common sequences that involve alternating differences like this?
3. Could this pattern be explained by a geometric or exponential sequence?
4. How might we generate the next set of integers after (+4, +1)?
5. Could a different set of operations (e.g., multiplication instead of addition) explain these transitions?

Tip: When looking for patterns, it’s helpful to check both arithmetic and geometric progressions for possibilities.