Which of the following numbers will replace the question mark (?) in the given series?
6, 8, 11, 16, ?, 34, 47, 64

To determine which number will replace the question mark (?) in the series, let's examine the pattern between the numbers.

Given series: 6, 8, 11, 16, ?, 34, 47, 64

First, let's find the differences between the consecutive terms:

8 - 6 = 2  
11 - 8 = 3  
16 - 11 = 5  
? - 16  
34 - ?  
47 - 34 = 13  
64 - 47 = 17

We can observe the differences: 2, 3, 5, ?, 13, 17. 

These differences do not form a straightforward arithmetic sequence, so let's check if they form a pattern of another type. Notably, the differences between the differences (second differences) might reveal a pattern:

3 - 2 = 1  
5 - 3 = 2  
? - 5  
13 - ?  
17 - 13 = 4

Looking at these second differences: 1, 2, ?, 4, we can see they increase by 1.

Following this pattern, the second difference that replaces the question mark should be 3 (since 1, 2, 3, 4).

Now, we find the missing first difference:

5 + 3 = 8

So, the first difference before 13 is 8. Hence, the missing term is:

16 + 8 = 24

Thus, the series with the missing number filled in is:
6, 8, 11, 16, 24, 34, 47, 64.

Therefore, the number that will replace the question mark is **24**.

Would you like further details or have any questions?

Here are 5 questions you might want to ask:
1. How do you identify patterns in number sequences?
2. What are arithmetic and geometric sequences?
3. How can second differences help identify patterns in sequences?
4. What are some common types of number series problems?
5. Can you provide another example of solving a number series problem?

Tip: When solving number sequences, always check for both first differences (differences between consecutive terms) and second differences (differences between first differences) to identify possible patterns.

Explore how to identify patterns in number sequences to solve the problem of finding the missing number in the series 6, 8, 11, 16, ?, 34, 47, 64. Learn about arithmetic patterns and second differences in sequences. Suitable for high school students.

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