https://file.aiquickdraw.com/mathbot/file/1727013164025w6ix1uxu.jpg

Let's go through the missing numbers in the sequences:

### a. \( 2, 4, \square, 16, 32, \dots \)
This seems to be a geometric sequence where each term doubles the previous one:
- \( 2 \times 2 = 4 \)
- \( 4 \times 2 = 8 \) (missing number)
- \( 8 \times 2 = 16 \)
- \( 16 \times 2 = 32 \)

So, the missing number is **8**.

### b. \( 100, 81, 64, \square, 36, \dots \)
This sequence shows a decreasing pattern. Let's find the differences:
- \( 100 - 81 = 19 \)
- \( 81 - 64 = 17 \)

It seems that the differences decrease by 2:
- The next difference should be \( 64 - \square = 15 \)

So, the missing number is **49**.

### c. \( 0, 15, 4, \square, 12, \dots \)
Here we need to observe how the numbers change:
- \( 15 - 0 = 15 \)
- \( 15 - 4 = 11 \)

We see a pattern where the difference decreases by 4, so:
- \( 4 - \square = 7 \)
- The missing number is **-3**.

### d. \( 1, 7, 17, \square, 49, \dots \)
This sequence looks like increasing squares or a quadratic relationship:
- \( 7 - 1 = 6 \)
- \( 17 - 7 = 10 \)

The difference increases by 4 each time. So the next difference should be \( 10 + 4 = 14 \):
- \( 17 + 14 = 31 \)

The missing number is **31**.

### e. \( \square, 6, 11, 16, 21, \dots \)
The sequence increases by 5 each time:
- \( 6 - \square = 5 \)

The missing number is **1**.

### Final answers:
a. **8**  
b. **49**  
c. **-3**  
d. **31**  
e. **1**

Would you like further details or have any questions?

Here are 5 related questions to consider:
1. How do you identify a geometric sequence?
2. What is the difference between arithmetic and geometric sequences?
3. How can quadratic relationships help in solving number sequences?
4. Why do the differences between terms in some sequences change at a constant rate?
5. How can negative numbers appear in sequences, and what does it signify?

**Tip**: When solving sequences, always check for common differences or ratios first—it can quickly reveal the pattern.

Fill in the missing numbers: 
a. 2, 4, __, 16, 32, ...
b. 100, 81, 64, __, 36, ...
c. 0, 15, 4, __, 12, ...
d. 1, 7, 17, __, 49, ...
e. __, 6, 11, 16, 21, ...

Solve missing number problems involving arithmetic and geometric sequences. This problem tests your knowledge of number patterns, suitable for Grades 6-8. Learn to apply formulas for both arithmetic and geometric sequences.

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