Understanding the Number Series Pattern

This question asks us to find the next number in the given series: 12, 2, 24, 3, 72, 4, ?. To solve this, we need to carefully examine the numbers and identify the underlying pattern or rule governing the sequence.

Let's look at the relationship between consecutive terms, but sometimes in number series, the pattern involves alternate terms or groups of terms. Observing the series, we see a mix of larger and smaller numbers, suggesting that there might be more than one pattern interleaved within the series.

Identifying the Interleaved Patterns

Let's separate the series into two sub-series based on their positions (odd and even):

Series: 12, 2, 24, 3, 72, 4, ?

• Terms at odd positions (1st, 3rd, 5th, 7th, ...): 12, 24, 72, ?
• Terms at even positions (2nd, 4th, 6th, ...): 2, 3, 4

Analyzing the Pattern in Odd Position Terms

Let's look at the sequence formed by the terms at the odd positions: 12, 24, 72, ?

• From 12 to 24: \( 12 \times 2 = 24 \)
• From 24 to 72: \( 24 \times 3 = 72 \)

It appears that each term in this sub-series is obtained by multiplying the previous term by a consecutive integer starting from 2. The multipliers are 2, 3, and the next multiplier should be 4.

So, the next term in the odd position sub-series would be \( 72 \times 4 \).

Analyzing the Pattern in Even Position Terms

Now let's look at the sequence formed by the terms at the even positions: 2, 3, 4.

This is a simple arithmetic progression where each term is obtained by adding 1 to the previous term.

• From 2 to 3: \( 2 + 1 = 3 \)
• From 3 to 4: \( 3 + 1 = 4 \)

The next term in this sub-series would be \( 4 + 1 = 5 \). However, the question mark is at the 7th position, which is an odd position, so we only need the next term in the odd-positioned sub-series.

Calculating the Missing Term

The missing term is the 7th term in the original series, which corresponds to the 4th term in the odd-positioned sub-series (1st, 3rd, 5th, 7th...). Following the pattern we identified for odd positions:

• 1st term: 12
• 3rd term: \( 12 \times 2 = 24 \)
• 5th term: \( 24 \times 3 = 72 \)
• 7th term: \( 72 \times 4 = 288 \)

Therefore, the number that replaces the question mark is 288.

Summary of the Pattern

The pattern for odd positions is multiplying the previous odd term by an increasing integer (2, 3, 4, ...).

The pattern for even positions is adding 1 to the previous even term (2, 3, 4, ...).

The next term in the series is at the 7th position, which follows the pattern for odd-positioned terms. So, the 7th term is \( 72 \times 4 = 288 \).

Conclusion

Based on the interleaved pattern observed in the series, the number that replaces the question mark is 288.

Revision Table: Number Series Patterns

Understanding different types of number series patterns is crucial for solving these problems. Here are some common types:

• Arithmetic Series: Constant difference between consecutive terms (e.g., 2, 4, 6, 8).
• Geometric Series: Constant ratio between consecutive terms (e.g., 3, 9, 27, 81).
• Difference Series: The difference between consecutive terms forms a pattern (e.g., 1, 2, 4, 7, 11... differences are 1, 2, 3, 4...).
• Product Series: Terms are derived by multiplying previous terms or applying a multiplication pattern (like the odd terms in this problem).
• Square/Cube Series: Terms are squares or cubes of natural numbers or follow a pattern related to squares/cubes (e.g., 1, 4, 9, 16 or 1, 8, 27, 64).
• Alternating Series: The pattern alternates between two different operations or involves two interleaved sub-series (like in this problem).
• Fibonacci Series: Each term is the sum of the two preceding terms (e.g., 0, 1, 1, 2, 3, 5, 8).

Additional Information: Tips for Solving Number Series Questions

Solving number series questions requires practice and a systematic approach. Here are some tips:

• Look for simple arithmetic (+, -, *, /) operations between consecutive terms.
• Check for patterns in the differences between consecutive terms.
• Examine the ratio between consecutive terms for geometric series.
• Consider squares, cubes, or other powers of numbers.
• Look for alternating patterns or interleaved series, as seen in this problem. Separate the terms at odd and even positions and look for patterns in each sub-series.
• If the numbers are changing rapidly (increasing or decreasing), multiplication, division, or squares/cubes might be involved.
• If the numbers are changing slowly, addition or subtraction might be involved.
• Don't give up easily; try different approaches if the first one doesn't reveal a clear pattern.