. Find the missing value in the series: 51, 52, 60, 87, 151, _____________, 492.

195
276
317
420

To find the missing value in the series, let's examine the pattern between the numbers:

1. **Difference between 52 and 51**:  
   \( 52 - 51 = 1 \)

2. **Difference between 60 and 52**:  
   \( 60 - 52 = 8 \)

3. **Difference between 87 and 60**:  
   \( 87 - 60 = 27 \)

4. **Difference between 151 and 87**:  
   \( 151 - 87 = 64 \)

These differences suggest a pattern related to perfect squares:
- \( 1 = 1^2 \)
- \( 8 = 2^3 \)
- \( 27 = 3^3 \)
- \( 64 = 4^3 \)

Next, the differences seem to follow cubes of integers:
- The next difference should be \( 5^3 = 125 \).

So, the missing value is:
\[ 151 + 125 = 276 \]

Finally, let's confirm this:
- The difference between 276 and 492:
  \( 492 - 276 = 216 \), which is \( 6^3 \).

The missing value in the series is **276**.

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Would you like more details or have any questions? Here are some related questions:

1. What is the next number in the series after 492?
2. How do patterns in number sequences help in determining missing values?
3. Can you identify other sequences that follow cubic patterns?
4. How would the pattern change if differences followed squares instead of cubes?
5. What are common methods for identifying patterns in number sequences?

**Tip:** When analyzing a sequence, consider the differences between consecutive terms. These differences often follow a pattern themselves, such as being constant, linear, or quadratic.

Find the missing value in the series: 51, 52, 60, 87, 151, _____________, 492.
Options: 195, 276, 317, 420.

Discover how to find the missing value in the sequence 51, 52, 60, 87, 151, ____, 492 using cubic patterns. This problem explores the use of cubic numbers to identify differences in sequences. Ideal for students in Grades 9-12.

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