Let's analyze and solve the patterns presented in the image.

1. Pattern a: 3, 6, 9, ___ , ___

   • The difference between terms is +3.
   • The next two terms will be: $9 + 3 = 12$ and $12 + 3 = 15$.
   • So, the sequence becomes: 3, 6, 9, 12, 15.

2. Pattern b: 3, 6, 12, ___ , ___

   • The first difference is +3, and then +6.
   • This seems to follow a doubling pattern for each step.
   • The next difference will be $+12$ and $+24$.
   • So, $12 + 12 = 24$ and $24 + 24 = 48$.
   • The sequence becomes: 3, 6, 12, 24, 48.

3. Pattern c: 1, 9, 13, ___ , ___

   • The difference between the first two terms is $+8$, and between the second and third is $+4$.
   • The difference appears to be decreasing by 4 each time.
   • The next difference would be $+2$, so $13 + 2 = 15$.
   • Then, the difference would be $+1$, so $15 + 1 = 16$.
   • The sequence becomes: 1, 9, 13, 15, 16.

4. Pattern d: 128, 64, ___ , ___ , 8

   • The pattern appears to be dividing by 2.
   • $128 \div 2 = 64$, and $64 \div 2 = 32$.
   • Continuing this pattern: $32 \div 2 = 16$ and $16 \div 2 = 8$.
   • The sequence becomes: 128, 64, 32, 16, 8.

Final Answer:

• a. 12, 15
• b. 24, 48
• c. 15, 16
• d. 32, 16

Would you like more details on any of these patterns?

Here are 5 related questions to think about:

1. What would the 6th term in sequence b be?
2. Can you create your own pattern using subtraction?
3. How would the sequences change if we added a constant instead of increasing it each time?
4. What happens to pattern d if we change the division to multiplication?
5. Can you identify any other number sequences that might fit these patterns?

Tip: Identifying differences between terms is often the key to finding the rule behind a sequence!