Extend the following patterns: a) 1, 2, 3, 5, 7, 11, , , b) 1, 3, 7, 15, , , c) 100, 52, 28, , , ___

The prime numbers are numbers greater than 1 that have no positive divisors other than 1 and themselves.
The prime numbers after 11 are 13, 17, and 19.

The extended pattern is: $1, 2, 3, 5, 7, 11, \boxed{\text{13}}, \boxed{\text{17}}, \boxed{\text{19}}$.

b) Pattern: 1, 3, 7, 15, \text{___}, \text{___}, \text{___}
Rule: multiply by 2 and add 1.

$$63 \times 2 + 1 = 126 + 1 = 127$$

The extended pattern is: $1, 3, 7, 15, \boxed{\text{31}}, \boxed{\text{63}}, \boxed{\text{127}}$.

c) Pattern: 100, 52, 28, \text{___}, \text{___}, \text{___}
Rule: divide by 2 and add 2.

$$10 \div 2 + 2 = 5 + 2 = 7$$

The extended pattern is: $100, 52, 28, \boxed{\text{16}}, \boxed{\text{10}}, \boxed{\text{7}}$.

3. Create two of your own number patterns and ask your friend to extend it.

a) Pattern: 5, 10, 15, 20, \text{___}, \text{___}, \text{___}
Rule: Add 5 to the previous number.

b) Pattern: 1, 4, 9, 16, \text{___}, \text{___}, \text{___}
Rule: The $n$-th term in the sequence is $n^2$.