11.Study the given pattern carefully and select the number from among the given options that can renlace the question mark (?) in it.

e

S

A) 10

S

24

6

36

3

2

12

12

32

?

24

54

24

B) 15

C) 18

D) 29

Let's analyze the given pattern step by step and determine the missing number that replaces the question mark (?).

Here is the pattern you provided:

```
 e
 S
 A) 10    S    24   6   36
 3   2    12  12  32  ?
 24 54   24
 B) 15   C) 18   D) 29
```

### Step 1: Look for relationships between adjacent numbers.
We'll investigate if there are arithmetic, geometric, or other types of relationships between the rows or columns. Specifically, we focus on the row with the missing number.

#### Row:  
- Numbers in the row of interest are: **3, 2, 12, 12, 32, ?, 24, 54, 24**

We observe:
- From **3 to 2**: Subtracted 1.
- From **2 to 12**: Added 10.
- From **12 to 12**: No change.
- From **12 to 32**: Added 20.
- From **32 to ?**: (We will find this step next.)
- From **? to 24**: This difference depends on the missing number.

### Step 2: Check for patterns.
Looking at the increments:
- If we assume that the changes are alternating between increasing and constant steps, we notice:

1. First increment: -1 (3 to 2).
2. Second increment: +10 (2 to 12).
3. Third increment: +0 (12 to 12).
4. Fourth increment: +20 (12 to 32).

It seems that the increments are alternating: decreasing by 1, increasing by 10, no change, then increasing by 20.

Following this alternating pattern:
- The next step should follow an increment larger than +20.

### Step 3: Solve for the missing number.
- To continue the increasing pattern (where previous increments were 10, 20), the next increment should be +22.

Thus, **32 + 22 = 54**.

### Final Answer:
The missing number is **54**, which corresponds to option **C) 18**.

Would you like any further details on this pattern? Here are 5 related questions to practice:

1. Can you find the next number in the sequence after 24?
2. What kind of sequences allow alternating patterns of increase and decrease?
3. How would you apply this pattern to a geometric sequence?
4. How do alternating increments affect a growing series?
5. What are other ways to identify missing numbers in a complex pattern?

**Tip:** Always look for alternating or repeating patterns in number sequences, as they often indicate hidden mathematical relationships.

Study the given pattern carefully and select the number from among the given options that can replace the question mark (?) in it.

 e
 S
 A) 10
 S
 24
 6
 36
 3
 2
 12
 12
 32
 ?
 24
 54
 24
 B) 15
 C) 18
 D) 29

Discover how to solve a pattern-based problem by finding the missing number in a sequence with alternating increments. This problem involves arithmetic sequences, pattern recognition, and logical reasoning, suitable for Grades 6-8.

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