Solving the Missing Number in Series: 2, 12, 30, ?, 90, 132

This question asks us to find the number that should replace the question mark (?) in the given number series: 2, 12, 30, ?, 90, 132.

To solve number series problems, we look for a pattern in the sequence of numbers. Let's examine the differences between consecutive terms.

Identifying the Pattern in the Number Series

Calculate the difference between each consecutive pair of numbers:

• Difference between the 2nd and 1st term: \(12 - 2 = 10\)
• Difference between the 3rd and 2nd term: \(30 - 12 = 18\)
• Difference between the last and 5th term: \(132 - 90 = 42\)

So far, the differences are 10, 18, ?, ?, 42. Let's look at the differences between these differences (the second differences).

• Difference between the 2nd and 1st difference: \(18 - 10 = 8\)

It appears there might be a constant second difference of 8. Let's assume this pattern continues.

Applying the Second Difference Pattern

If the second difference is constant at 8, then the next difference after 18 should be \(18 + 8 = 26\). This difference is between the 3rd term (30) and the missing term (?).

So, the missing term is \(30 + 26 = 56\).

Verifying the Pattern with the Calculated Missing Number

Let's insert 56 into the series: 2, 12, 30, 56, 90, 132.

Now, calculate the differences again:

• \(12 - 2 = 10\)
• \(30 - 12 = 18\)
• \(56 - 30 = 26\)
• \(90 - 56 = 34\)
• \(132 - 90 = 42\)

The first differences are: 10, 18, 26, 34, 42.

Now, calculate the second differences from these first differences:

• \(18 - 10 = 8\)
• \(26 - 18 = 8\)
• \(34 - 26 = 8\)
• \(42 - 34 = 8\)

The second differences are indeed constant at 8. This confirms that 56 is the correct number to replace the question mark.

Alternative Pattern Recognition

Another way to observe the pattern is by looking at the terms as products of consecutive numbers:

• \(2 = 1 \times 2\)
• \(12 = 3 \times 4\)
• \(30 = 5 \times 6\)
• The next term should follow this pattern: \(7 \times 8 = 56\)
• \(90 = 9 \times 10\)
• \(132 = 11 \times 12\)

This pattern also confirms that the missing number is 56. The series terms are products of consecutive odd and even numbers: (1x2), (3x4), (5x6), (7x8), (9x10), (11x12).

Conclusion

Based on the pattern of a constant second difference of 8, or the pattern of products of consecutive odd and even numbers, the number that replaces the question mark in the series 2, 12, 30, ?, 90, 132 is 56.

Number Series Revision Table

Additional Information on Number Series Patterns

Number series questions are common in aptitude tests. They assess logical reasoning and pattern recognition skills. Common types of patterns include:

• Arithmetic Series: A constant difference between consecutive terms.
• Geometric Series: A constant ratio between consecutive terms.
• Arithmetic-Geometric Series: A combination of arithmetic and geometric progression.
• Difference Series: Examining the differences between terms (as done in this problem). This can lead to finding patterns in the first differences, second differences, and so on.
• Product/Division Series: Terms related by multiplication or division.
• Square/Cube Series: Terms based on squares or cubes of numbers, possibly with additions or subtractions.
• Fibonacci Series: Each term is the sum of the two preceding terms (e.g., 1, 1, 2, 3, 5, 8...).
• Mixed Series: Multiple patterns combined or alternating.

Identifying the pattern often involves calculating differences, ratios, or looking for common mathematical operations between terms.