To determine the next three numbers in each pattern, let's analyze them individually.

Pattern 1: $7, 9, 14$

• From $7$ to $9$, the difference is $9 - 7 = 2$.
• From $9$ to $14$, the difference is $14 - 9 = 5$.

The differences are $+2$ and $+5$. It appears that the differences are increasing by $3$. So, the next difference should be $5 + 3 = 8$.

• The next number would be $14 + 8 = 22$.

For the following number, the difference should increase by another $3$, i.e., $8 + 3 = 11$.

• The next number would be $22 + 11 = 33$.

The next difference will be $11 + 3 = 14$.

• The next number would be $33 + 14 = 47$.

So, the next three numbers in the first pattern are: 22, 33, 47.

Pattern 2: $7, 13, 18$

• From $7$ to $13$, the difference is $13 - 7 = 6$.
• From $13$ to $18$, the difference is $18 - 13 = 5$.

The differences are decreasing by $1$, from $+6$ to $+5$. If the differences continue decreasing by $1$, the next difference should be $5 - 1 = 4$.

• The next number would be $18 + 4 = 22$.

The next difference should be $4 - 1 = 3$.

• The next number would be $22 + 3 = 25$.

The next difference should be $3 - 1 = 2$.

• The next number would be $25 + 2 = 27$.

So, the next three numbers in the second pattern are: 22, 25, 27.

Pattern 3: $1, 2, 6$

• From $1$ to $2$, the difference is $2 - 1 = 1$.
• From $2$ to $6$, the difference is $6 - 2 = 4$.

The differences are $+1$ and $+4$. The differences increased by $3$. If the difference continues increasing by $3$, the next difference should be $4 + 3 = 7$.

• The next number would be $6 + 7 = 13$.

The next difference should increase by $3$, so it becomes $7 + 3 = 10$.

• The next number would be $13 + 10 = 23$.

The next difference should increase by $3$, so it becomes $10 + 3 = 13$.

• The next number would be $23 + 13 = 36$.

So, the next three numbers in the third pattern are: 13, 23, 36.

Summary of the next three numbers in each pattern:

• First pattern: 22, 33, 47
• Second pattern: 22, 25, 27
• Third pattern: 13, 23, 36

Would you like more details on any of these patterns or have further questions?

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Here are 5 related questions:

1. How can we identify the general rule for an arithmetic sequence?
2. What is the difference between arithmetic and geometric sequences?
3. Can differences in a pattern always be constant, or can they follow a formula?
4. What are the characteristics of a quadratic number sequence?
5. How can you find the $n$-th term of a pattern if given a few terms?

Tip: If the differences in a sequence aren't constant, check if they are increasing or decreasing by a set number. This might indicate a second-level pattern.