Understanding Number Series Patterns

A number series question requires identifying a specific pattern or rule that governs the sequence of numbers. Once the pattern is found, it can be used to find the missing number in the series.

Analyzing the Given Number Sequence

The given sequence of numbers is:

24, 18, 12, 6, 14, 9, 8, 7, ?, 72, 11, 6, 48

Let's examine the relationships between adjacent numbers and look for potential patterns. Sometimes, a single pattern applies throughout the series, while other times, multiple patterns might be interwoven.

Discovering Interwoven Sequences

Upon careful observation, this series appears to be composed of multiple independent sequences interwoven together based on their position in the original series. Let's categorize the numbers based on their index (position) in the sequence, starting from index 1.

• Numbers at indices 1, 5, 9, 13: 24, 14, ?, 48
• Numbers at indices 2, 6, 10: 18, 9, 72
• Numbers at indices 3, 7, 11: 12, 8, 11
• Numbers at indices 4, 8, 12: 6, 7, 6

Pattern for Each Interwoven Sequence

Let's analyze the pattern within each of these derived sequences:

Sequence 1 (Indices 1, 5, 9, 13: 24, 14, ?, 48)

Look at the operations between consecutive terms:

• From 24 to 14: \(24 - 10 = 14\)
• From 14 to ?: Let's assume the pattern continues. If the operation is \(-10\), then \(14 - 10 = 4\). Let's check if this fits the next step.
• From ? (assuming 4) to 48: \(4 \times 12 = 48\)

The pattern for this sequence seems to be: Subtract 10, Subtract 10, Multiply by 12.

Sequence 1: \(24 \xrightarrow{-10} 14 \xrightarrow{-10} 4 \xrightarrow{\times 12} 48\)

This pattern holds for the known numbers if the missing number is 4.

Sequence 2 (Indices 2, 6, 10: 18, 9, 72)

Look at the operations:

• From 18 to 9: \(18 \div 2 = 9\)
• From 9 to 72: \(9 \times 8 = 72\)

The pattern for this sequence is: Divide by 2, Multiply by 8.

Sequence 2: \(18 \xrightarrow{\div 2} 9 \xrightarrow{\times 8} 72\)

Sequence 3 (Indices 3, 7, 11: 12, 8, 11)

Look at the operations:

• From 12 to 8: \(12 - 4 = 8\)
• From 8 to 11: \(8 + 3 = 11\)

The pattern for this sequence is: Subtract 4, Add 3.

Sequence 3: \(12 \xrightarrow{-4} 8 \xrightarrow{+3} 11\)

Sequence 4 (Indices 4, 8, 12: 6, 7, 6)

Look at the operations:

• From 6 to 7: \(6 + 1 = 7\)
• From 7 to 6: \(7 - 1 = 6\)

The pattern for this sequence is: Add 1, Subtract 1.

Sequence 4: \(6 \xrightarrow{+1} 7 \xrightarrow{-1} 6\)

Calculating the Missing Number

The question mark (?) is at index 9 in the original series. This position belongs to Sequence 1 (Indices 1, 5, 9, 13).

Sequence 1 is 24, 14, ?, 48.

The pattern for Sequence 1 is Subtract 10, Subtract 10, Multiply by 12.

The number at index 5 is 14. To find the number at index 9 (the question mark), we apply the second operation in the pattern to the number at index 5.

Second operation for Sequence 1 is Subtract 10.

\(? = 14 - 10 = 4\)

So, the missing number is 4.

Verifying the Pattern

If the missing number is 4, the sequence is:

24, 18, 12, 6, 14, 9, 8, 7, 4, 72, 11, 6, 48

Let's check the interwoven sequences again:

• Indices 1, 5, 9, 13: 24, 14, 4, 48. Pattern: \(24-10=14\), \(14-10=4\), \(4 \times 12=48\). (Consistent)
• Indices 2, 6, 10: 18, 9, 72. Pattern: \(18 \div 2=9\), \(9 \times 8=72\). (Consistent)
• Indices 3, 7, 11: 12, 8, 11. Pattern: \(12-4=8\), \(8+3=11\). (Consistent)
• Indices 4, 8, 12: 6, 7, 6. Pattern: \(6+1=7\), \(7-1=6\). (Consistent)

The pattern holds true with 4 as the missing number.

Final Answer

The number that comes in place of the question mark (?) is 4.

Comparing this with the given options:

Option 1: 6

Option 2: 3

Option 3: 4

Option 4: 9

The calculated missing number matches Option 3.