Finding the Missing Number in a Series

Let's analyze the given number series to find the pattern and determine the number that replaces the question mark.

The series is: 18, 3, 36, 6, 54, 9, ?

Observing the series, we can see that the numbers seem to alternate between a larger value and a smaller value. This often indicates an interleaved or alternating pattern where two separate series are combined.

Let's separate the numbers at odd positions and the numbers at even positions:

• Numbers at odd positions (1st, 3rd, 5th, 7th): 18, 36, 54, ?
• Numbers at even positions (2nd, 4th, 6th): 3, 6, 9

Analyzing the Odd Positions Series

The series at odd positions is 18, 36, 54, ?. Let's look for a pattern:

• From 18 to 36: \(36 - 18 = 18\). Alternatively, \(18 \times 2 = 36\) is incorrect for the next step. Let's check addition.
• From 36 to 54: \(54 - 36 = 18\).

It appears that a constant value of 18 is added to each term to get the next term in this sub-series. This is an arithmetic progression with a common difference of 18.

So, the next number in this odd positions series would be the last term plus 18:

Next number = \(54 + 18\)

Next number = \(72\)

Analyzing the Even Positions Series

The series at even positions is 3, 6, 9. Let's look for a pattern:

• From 3 to 6: \(6 - 3 = 3\). Alternatively, \(3 \times 2 = 6\). Let's check the next step.
• From 6 to 9: \(9 - 6 = 3\).

It appears that a constant value of 3 is added to each term to get the next term in this sub-series. This is an arithmetic progression with a common difference of 3.

So, the next number in this even positions series would be the last term plus 3:

Next number = \(9 + 3\)

Next number = \(12\)

Identifying the Position of the Question Mark

The question mark (?) is the seventh term in the original series. The seventh position is an odd position.

Therefore, the number replacing the question mark should follow the pattern of the odd positions series.

Calculating the Missing Number

Using the pattern for the odd positions series (adding 18 to the previous term):

The odd positions are: 1st (18), 3rd (36), 5th (54), 7th (?).

The number at the 7th position is the number at the 5th position plus 18.

Missing Number = \(54 + 18 = 72\)

Conclusion

The number that replaces the question mark in the series is 72.

The number 72 fits the identified pattern for the odd-positioned terms in the series.

Revision Table: Number Series Analysis

Additional Information: Types of Number Series Patterns

Number series problems test your logical reasoning and pattern recognition skills. Besides interleaved series and arithmetic progressions, other common patterns include:

• Geometric Progression: Each term is found by multiplying the previous term by a constant ratio (e.g., 2, 4, 8, 16...).
• Difference Series: The difference between consecutive terms follows a pattern (e.g., increasing differences, differences in AP or GP).
• Square or Cube Series: Terms are squares or cubes of natural numbers, or related to them (e.g., \(1^2, 2^2, 3^2, ...\) or \(n^2 \pm x\)).
• Fibonacci Series: Each term is the sum of the two preceding terms (e.g., 0, 1, 1, 2, 3, 5, 8...).
• Mixed Series: Patterns involving a combination of operations (e.g., \(\times 2 + 1\), \(- 3 \times 2\)).
• Alternating Operations: The operation alternates between terms (e.g., +5, -2, +5, -2...).

Solving number series questions requires careful observation and testing different types of patterns to find the correct one that fits the entire sequence.