Understanding the Number Series Problem

The problem asks us to find the next number in the given series: 4, 8, 12, 24, 18, ?. To solve this, we need to identify the logical pattern or rule that connects the numbers in the sequence.

Identifying the Number Series Pattern

Let's look at the relationship between consecutive terms in the series:

• From 4 to 8: The operation is multiplication by 2 (\(4 \times 2 = 8\)).
• From 8 to 12: The operation is addition of 4 (\(8 + 4 = 12\)).
• From 12 to 24: The operation is multiplication by 2 (\(12 \times 2 = 24\)).
• From 24 to 18: The operation is subtraction of 6 (\(24 - 6 = 18\)).
• From 18 to ?: We need to find the operation.

Let's list the operations found so far: \(\times 2, +4, \times 2, -6\).

We can observe that the operation \(\times 2\) appears to alternate with other operations. Let's look at the operations that are not \(\times 2\): \(+4\) and \(-6\). The difference between these two values is \(-6 - (+4) = -10\).

This suggests a pattern where the operations alternate between multiplication by 2 and an additive/subtractive value that decreases by 10 each time it appears.

Step-by-Step Solution to Complete the Series

Based on the identified pattern (\(\times 2\), then Add/Subtract, then \(\times 2\), then Add/Subtract, and so on):

1. The operation from the 1st term (4) to the 2nd term (8) is \(\times 2\).
2. The operation from the 2nd term (8) to the 3rd term (12) is \(+4\).
3. The operation from the 3rd term (12) to the 4th term (24) is \(\times 2\).
4. The operation from the 4th term (24) to the 5th term (18) is \(-6\).

The sequence of operations is: \(\times 2, +4, \times 2, -6\). Following the pattern of alternating \(\times 2\) and then the changing additive/subtractive operation:

• The operation after \(\times 2\) was \(+4\).
• The operation after the next \(\times 2\) was \(-6\).
• The next operation in the overall sequence should be \(\times 2\).

So, to find the next term after 18, we apply the \(\times 2\) operation to the last term (18).

\(18 \times 2 = 36\)

The next number in the series is 36.

The complete series is 4, 8, 12, 24, 18, 36.

Understanding the Pattern Sequence

The sequence of operations applied to get the next term is:

• Term 1 \(\xrightarrow{\times 2}\) Term 2
• Term 2 \(\xrightarrow{+4}\) Term 3
• Term 3 \(\xrightarrow{\times 2}\) Term 4
• Term 4 \(\xrightarrow{-6}\) Term 5
• Term 5 \(\xrightarrow{\times 2}\) Term 6

The additive/subtractive operations are +4, -6. The next such operation would be -16 (as +4 to -6 is a decrease of 10), but this would occur after the *next* \(\times 2\) operation, meaning it would apply from Term 6 to Term 7.

The pattern of operations is \(\times 2, +4, \times 2, -6, \times 2, \ldots\). The next operation is \(\times 2\).

Therefore, the missing term is \(18 \times 2 = 36\).

Revision Table: Number Series Patterns

Additional Information: Solving Number Series Questions

Solving number series questions often involves looking for different types of patterns. Here are some strategies:

• Look at differences: Calculate the difference between consecutive terms. See if these differences form a recognizable pattern.
• Look at ratios: Calculate the ratio between consecutive terms (divide a term by the previous one). See if these ratios form a pattern.
• Check for alternating patterns: See if operations or patterns alternate between terms. This can be simple (e.g., +2, -1, +2, -1) or more complex as seen in this question.
• Consider interleaved series: Sometimes, the series is a combination of two or more simpler series placed one after another.
• Look for squares, cubes, prime numbers, etc.: The pattern might involve these special numbers or operations related to them.
• Combine operations: The pattern might involve a combination of addition, subtraction, multiplication, division, or even exponents.
• Position-based patterns: The pattern might depend on the position of the term in the series (e.g., add the position number).

Practicing various types of series helps in quickly identifying the underlying pattern.