Solving the Number Series: 12, 36, ?, 102, 100, 300

This question asks us to find the number that replaces the question mark (?) in the given series: 12, 36, ?, 102, 100, 300. To solve this, we need to identify the pattern or rule governing the sequence of numbers.

Analyzing the Number Series Pattern

Let's look at the relationship between consecutive numbers and numbers further apart in the series:

• From 12 to 36: \(12 \times 3 = 36\). This suggests a multiplication by 3.
• From 102 to 100: \(102 - 2 = 100\). This suggests a subtraction by 2.
• From 100 to 300: \(100 \times 3 = 300\). This again suggests a multiplication by 3.

Based on these observations, a possible pattern emerges: the series alternates between multiplying the previous term by 3 and subtracting 2 from the previous term.

Applying the Pattern to Find the Missing Number

Let's test this alternating pattern on the series starting from the first term:

• Term 1: 12
• Applying the first operation (multiply by 3) to Term 1: \(12 \times 3 = 36\). This matches Term 2.
• Applying the second operation (subtract 2) to Term 2: \(36 - 2 = 34\). This should be Term 3 (the missing number).
• Applying the first operation (multiply by 3) to Term 3: \(34 \times 3 = 102\). This matches Term 4.
• Applying the second operation (subtract 2) to Term 4: \(102 - 2 = 100\). This matches Term 5.
• Applying the first operation (multiply by 3) to Term 5: \(100 \times 3 = 300\). This matches Term 6.

The pattern holds true for the entire series if we assume the missing number is 34.

The pattern can be formally described as:

• If the term number \(n\) is odd, the next term (\(T_{n+1}\)) is \(T_n \times 3\).
• If the term number \(n\) is even, the next term (\(T_{n+1}\)) is \(T_n - 2\).

Let's verify again:

• \(T_1 = 12\) (n=1, odd)
• \(T_2 = T_1 \times 3 = 12 \times 3 = 36\) (n=2, even)
• \(T_3 = T_2 - 2 = 36 - 2 = 34\) (n=3, odd)
• \(T_4 = T_3 \times 3 = 34 \times 3 = 102\) (n=4, even)
• \(T_5 = T_4 - 2 = 102 - 2 = 100\) (n=5, odd)
• \(T_6 = T_5 \times 3 = 100 \times 3 = 300\)

The missing term is indeed 34.

Conclusion

The missing number in the series 12, 36, ?, 102, 100, 300 is 34, following the alternating pattern of multiplying by 3 and subtracting 2.

Revision Table: Key Concepts for Number Series

Additional Information on Solving Number Series Problems

Solving number series questions is a common part of logical reasoning and quantitative aptitude tests. Here are some tips:

• Look for simple arithmetic progressions (addition, subtraction).
• Check for geometric progressions (multiplication, division).
• Examine the differences between consecutive terms. Sometimes the differences form a simpler series.
• Look for alternating patterns involving different operations or sequences.
• Consider squares, cubes, prime numbers, or Fibonacci sequences if other patterns aren't obvious.
• Practice with various types of number series to become familiar with common patterns.
• Write down the terms and the differences or ratios between them to help spot the pattern.

Understanding these basic types and techniques will help you approach different number series problems systematically and efficiently.