Solving the Number Series Pattern Question

The question asks us to identify the pattern in the given series and find the missing number. The series is: 1, 5, 3, 2, 10, 6, 4, 15, 9, 8, ?, 12.

Let's examine the numbers to find a logical sequence or pattern. Observing the series, it appears to be composed of multiple interleaved patterns. Let's try grouping the numbers into sets of three:

• Group 1: 1, 5, 3
• Group 2: 2, 10, 6
• Group 3: 4, 15, 9
• Group 4: 8, ?, 12

Now, let's look at the numbers in the same position within each group:

1. The first numbers of each group form a series: 1, 2, 4, 8.
2. The second numbers of each group form a series: 5, 10, 15, ?.
3. The third numbers of each group form a series: 3, 6, 9, 12.

Analyzing the Individual Series

Let's analyze the pattern for each of these new series:

• Series 1 (First terms): 1, 2, 4, 8

  This is a geometric progression where each term is obtained by multiplying the previous term by 2.

  \(1 \times 2 = 2\)

  \(2 \times 2 = 4\)

  \(4 \times 2 = 8\)

  This pattern is consistent.

• Series 2 (Second terms): 5, 10, 15, ?

  This is an arithmetic progression where a constant value is added to the previous term to get the next term. The difference between consecutive terms is \(10 - 5 = 5\) and \(15 - 10 = 5\). The common difference is 5.

  \(5 + 5 = 10\)

  \(10 + 5 = 15\)

  To find the next term (which is the missing number), we add 5 to the last known term in this series:

  \(15 + 5 = 20\)

  So, the missing number should be 20.

• Series 3 (Third terms): 3, 6, 9, 12

  This is also an arithmetic progression. The difference between consecutive terms is \(6 - 3 = 3\), \(9 - 6 = 3\), and \(12 - 9 = 3\). The common difference is 3.

  \(3 + 3 = 6\)

  \(6 + 3 = 9\)

  \(9 + 3 = 12\)

  This pattern is consistent and accounts for the last number in the original series.

Determining the Missing Number

The question mark is the 11th term in the original series. In our grouping of three terms, the 11th term falls into the 4th group (8, ?, 12) and is the second term of this group.

Therefore, the missing number belongs to the series formed by the second terms of the groups: 5, 10, 15, ?. Following the arithmetic progression pattern for this series (adding 5 each time), the next term after 15 is \(15 + 5 = 20\).

So, the missing number that replaces the question mark is 20.

Conclusion on the Number Series

Based on the identified patterns within the interleaved series, the missing number is 20.

Revision Table: Number Series Patterns

Reviewing the concept of number series patterns is crucial for solving such problems. Common types include:

• Arithmetic Progression (constant difference)
• Geometric Progression (constant ratio)
• Mixed Series (combination of arithmetic and geometric operations)
• Interleaved Series (multiple independent series combined)
• Differences Series (pattern in the differences between terms)

Additional Information: Logical Reasoning Skills

Solving number series questions enhances logical reasoning and analytical skills. Practice with various types of series helps in quickly identifying the underlying rule. It's often helpful to look for patterns in differences, ratios, or positions within the sequence, especially in complex or longer series.