Solving the Interleaved Number Series

The given series is 1, 3, 3, 6, 5, 12, 7, 24, 9, 48, 11, ____. To find the missing term, we first look for a pattern in the sequence. Often, when a pattern isn't immediately obvious across consecutive terms, the series might be composed of two separate series interleaved together.

Let's separate the terms based on their position in the sequence:

• Terms at odd positions (1st, 3rd, 5th, 7th, 9th, 11th): 1, 3, 5, 7, 9, 11
• Terms at even positions (2nd, 4th, 6th, 8th, 10th): 3, 6, 12, 24, 48

Analyzing the Odd-Position Series

The series formed by terms at odd positions is 1, 3, 5, 7, 9, 11. Let's examine the difference between consecutive terms in this series:

• \(3 - 1 = 2\)
• \(5 - 3 = 2\)
• \(7 - 5 = 2\)
• \(9 - 7 = 2\)
• \(11 - 9 = 2\)

The difference is constant and equal to 2. This indicates that the odd-position series is an arithmetic progression with a common difference of 2.

Analyzing the Even-Position Series

The series formed by terms at even positions is 3, 6, 12, 24, 48. Let's examine the relationship between consecutive terms in this series:

• \(6 \div 3 = 2\)
• \(12 \div 6 = 2\)
• \(24 \div 12 = 2\)
• \(48 \div 24 = 2\)

The ratio between consecutive terms is constant and equal to 2. This indicates that the even-position series is a geometric progression with a common ratio of 2.

Finding the Missing Term

The blank in the original series is after the 11th term. This position is the 12th term, which is an even position. Therefore, the missing term belongs to the even-position series (3, 6, 12, 24, 48, ____).

To find the next term in the even-position series, we apply the pattern observed: multiply the last term by the common ratio 2.

The last term in the observed even-position series is 48.

The next term is \(48 \times 2 = 96\).

So, the missing term in the series is 96.

Summary of the Series Patterns

Here is a table showing the two interleaved series and their patterns:

Revision Table: Key Concepts

Additional Information: Types of Number Series

Understanding different types of number series is crucial for solving reasoning questions. Besides arithmetic and geometric progressions, other patterns you might encounter include:

• Difference Series: The difference between consecutive terms follows a pattern (e.g., differences are 1, 2, 3, 4...).
• Ratio Series: The ratio between consecutive terms follows a pattern.
• Square/Cube Series: Terms are squares or cubes of natural numbers, or related to them.
• Fibonacci Series: Each term is the sum of the two preceding terms (e.g., 0, 1, 1, 2, 3, 5, 8...).
• Mixed Series: A combination of different patterns (like our interleaved series example) or where the pattern involves multiple operations.
• Alternating Series: The pattern alternates between two different operations (e.g., +2, -1, +2, -1...).

Solving number series questions requires careful observation and systematic analysis to identify the rule connecting the terms.