Understanding and Solving Number Series

Number series problems require identifying the underlying pattern or rule that governs the sequence of numbers. Once the pattern is found, we can use it to predict the next term or find a missing term.

Analyzing the Given Number Series Pattern

The number series provided is:

• 48
• 75
• 108
• ?
• 192

We need to determine the number that should replace the question mark.

Let's examine the relationship between the consecutive terms. We can try finding the differences between terms:

• \(75 - 48 = 27\)
• \(108 - 75 = 33\)

The differences are 27 and 33. The difference between these differences is \(33 - 27 = 6\). This suggests a pattern where the difference between consecutive terms increases by 6 each time.

Let's explore another potential pattern. Sometimes, number series involve operations like multiplication, squares, or cubes.

Let's look at the terms again:

• 48
• 75
• 108
• ?
• 192

Notice that these numbers might be related to squares. Let's see if we can factor out a common number or relate them to squares of integers:

• \(48 = 3 \times 16 = 3 \times 4^2\)
• \(75 = 3 \times 25 = 3 \times 5^2\)
• \(108 = 3 \times 36 = 3 \times 6^2\)

This reveals a clear pattern: each term is obtained by multiplying 3 by the square of a consecutive integer, starting from 4. The sequence of integers being squared is 4, 5, 6, ...

Calculating the Missing Term Using the Pattern

Following the identified pattern \(3 \times n^2\), the missing term is the fourth term in the series. The integers being squared are 4, 5, 6, so for the fourth term, the integer 'n' should be 7.

Let's calculate the fourth term:

Missing Term \( = 3 \times 7^2 \)

Missing Term \( = 3 \times 49 \)

Missing Term \( = 147 \)

Verifying the Pattern

To confirm the pattern, let's check the last term in the series (192). Following the pattern, the integer 'n' for the fifth term should be 8.

\(3 \times 8^2 = 3 \times 64 = 192\)

The last term fits the pattern perfectly. This confirms that the pattern \(3 \times n^2\) where \(n\) is 4, 5, 6, 7, 8 is correct for this series.

The Completed Series

Based on the pattern, the completed series is:

• 48
• 75
• 108
• 147
• 192

Conclusion

The number that replaces the question mark in the series is 147.

Revision Table: Common Number Series Types

Additional Information: Strategies for Solving Number Series

Approaching number series questions systematically can help. Here are some strategies:

• Calculate differences between consecutive terms. If there's no immediate pattern, calculate the differences between the differences (second-order difference).
• Calculate the ratio between consecutive terms. This is useful for geometric series.
• Check if terms relate to simple arithmetic operations (addition, subtraction, multiplication, division).
• Consider if terms are related to squares, cubes, prime numbers, or Fibonacci sequence.
• Look for alternating patterns involving two different rules applied to alternate terms.
• Practice recognizing common patterns to improve speed and accuracy.