Solving the Number Series Puzzle

Let's analyze the given number series to find the pattern and determine the missing term:

6, 6, 7, 7, 11, 15, 20, 42, 36, 106, 61, 231, ?, 447

This series appears to be an interleaved series, meaning it consists of two separate patterns alternating between positions.

Analyzing the Interleaved Number Series

We can split the given series into two sub-series:

Series 1: Terms at Odd Positions

These are the terms at the 1st, 3rd, 5th, 7th, 9th, 11th, 13th positions, and so on.

The terms are: 6, 7, 11, 20, 36, 61, ?

Let's look at the differences between consecutive terms in this sub-series:

• 7 - 6 = 1
• 11 - 7 = 4
• 20 - 11 = 9
• 36 - 20 = 16
• 61 - 36 = 25

The differences are 1, 4, 9, 16, 25. These are the squares of consecutive integers: \(1^2, 2^2, 3^2, 4^2, 5^2\).

Following this pattern, the next difference should be \(6^2 = 36\).

So, the next term in this series (which is the 13th term in the original series) should be 61 + 36.

61 + 36 = 97

Series 2: Terms at Even Positions

These are the terms at the 2nd, 4th, 6th, 8th, 10th, 12th, 14th positions, and so on.

The terms are: 6, 7, 15, 42, 106, 231, 447

Let's look at the differences between consecutive terms in this sub-series:

• 7 - 6 = 1
• 15 - 7 = 8
• 42 - 15 = 27
• 106 - 42 = 64
• 231 - 106 = 125
• 447 - 231 = 216

The differences are 1, 8, 27, 64, 125, 216. These are the cubes of consecutive integers: \(1^3, 2^3, 3^3, 4^3, 5^3, 6^3\). This confirms the interleaved pattern structure.

Calculating the Missing Term

The question mark (?) is at the 13th position in the original series. As we determined, the odd positions follow the pattern of adding consecutive squares.

The terms in the odd-position series are:

• 1st term: 6
• 3rd term: \(6 + 1^2 = 6 + 1 = 7\)
• 5th term: \(7 + 2^2 = 7 + 4 = 11\)
• 7th term: \(11 + 3^2 = 11 + 9 = 20\)
• 9th term: \(20 + 4^2 = 20 + 16 = 36\)
• 11th term: \(36 + 5^2 = 36 + 25 = 61\)
• 13th term: \(61 + 6^2 = 61 + 36 = 97\)

Therefore, the number that replaces the question mark is 97.

Revision Table: Number Series Patterns

Additional Information: Recognizing Number Series Patterns

To solve number series questions effectively, practice is key. Look for common patterns:

• Arithmetic Progression: Check if there's a fixed number added or subtracted.
• Geometric Progression: Check if there's a fixed number multiplied or divided.
• Differences: Calculate the difference between consecutive terms. Sometimes the differences themselves form a pattern (arithmetic, geometric, squares, cubes, etc.). You might need to calculate differences of differences.
• Ratio: Check the ratio between consecutive terms.
• Squares and Cubes: Look for numbers that are perfect squares or cubes, or if the pattern involves adding/subtracting squares or cubes.
• Alternating Patterns: Check if operations (like + and -) or patterns alternate.
• Interleaved Series: Separate the series into terms at odd positions and terms at even positions and analyze each sub-series independently. This is a common advanced pattern.
• Fibonacci-like Series: Each term is the sum of the previous two (or more) terms.

In this problem, identifying it as an interleaved series and then recognizing the square and cube patterns in the sub-series was crucial.