Analyzing the Number Series Pattern

The problem asks us to find the next number in the given series: 30, 38, 65, 129, 254, ?. To solve this, we need to identify the underlying pattern or rule that generates the terms of this number series.

Let's look at the differences between consecutive terms:

• Difference between the 2nd and 1st term: $38 - 30 = 8$
• Difference between the 3rd and 2nd term: $65 - 38 = 27$
• Difference between the 4th and 3rd term: $129 - 65 = 64$
• Difference between the 5th and 4th term: $254 - 129 = 125$

The sequence of differences is 8, 27, 64, 125.

Identifying the Pattern in Differences

Let's examine the sequence of differences (8, 27, 64, 125) more closely. We can observe that these numbers are perfect cubes:

• $8 = 2 \times 2 \times 2 = 2^3$
• $27 = 3 \times 3 \times 3 = 3^3$
• $64 = 4 \times 4 \times 4 = 4^3$
• $125 = 5 \times 5 \times 5 = 5^3$

The pattern in the differences is that they are the cubes of consecutive integers starting from 2 ($2^3, 3^3, 4^3, 5^3$).

Predicting the Next Term in the Series

Following this pattern, the next difference in the series should be the cube of the next consecutive integer after 5, which is 6.

• Next difference = $6^3 = 6 \times 6 \times 6 = 216$

To find the next term in the original series (which replaces the question mark), we add this next difference to the last known term (254).

• Next term = Last term + Next difference
• Next term = $254 + 216$
• Next term = $470$

Therefore, the number that replaces the question mark in the series is 470.

Conclusion

By analyzing the differences between consecutive terms, we found a pattern based on consecutive cubes. Applying this pattern, the next term in the series 30, 38, 65, 129, 254, ? is 470.

Revision Table - Number Series Analysis

Understanding number series requires looking for various patterns like differences, ratios, squares, cubes, prime numbers, or combinations of these. Let's quickly review the steps taken for this specific series.

Additional Information - Types of Number Series

Number series problems often involve recognizing different types of patterns. Here are a few common types:

• Arithmetic Series: Each term is obtained by adding a constant value (common difference) to the previous term (e.g., 2, 5, 8, 11,... difference is 3).
• Geometric Series: Each term is obtained by multiplying the previous term by a constant value (common ratio) (e.g., 3, 6, 12, 24,... ratio is 2).
• Difference Series: The differences between consecutive terms follow a pattern, which might be an arithmetic series, geometric series, squares, cubes, etc. (like in this problem).
• Squares/Cubes Series: Terms are related to squares or cubes of natural numbers (e.g., 1, 4, 9, 16,... are $1^2, 2^2, 3^2, 4^2$).
• Mixed Series: The pattern might involve a combination of operations (e.g., multiply by a number and add/subtract another number, or alternating operations).
• Fibonacci Series: Each term is the sum of the two preceding terms (e.g., 0, 1, 1, 2, 3, 5, 8,...).

Solving number series problems effectively requires practice in identifying these various patterns.