Understanding the Number Series Pattern

The question asks us to identify the pattern in the given number series and find the next number. The series is 3, 12, 30, 57, 93, 138, 192, ?. To solve this number series problem, we need to look at the differences between consecutive terms.

Analyzing the Differences in the Number Sequence

Let's calculate the differences between each pair of consecutive numbers in the series:

• Difference between the 2nd and 1st term: $12 - 3 = 9$
• Difference between the 3rd and 2nd term: $30 - 12 = 18$
• Difference between the 4th and 3rd term: $57 - 30 = 27$
• Difference between the 5th and 4th term: $93 - 57 = 36$
• Difference between the 6th and 5th term: $138 - 93 = 45$
• Difference between the 7th and 6th term: $192 - 138 = 54$

The sequence of these first differences is 9, 18, 27, 36, 45, 54.

Identifying the Pattern in the Differences

Now, let's examine the sequence of differences (9, 18, 27, 36, 45, 54). We can see if there is a pattern in these differences (a second level of differences).

• Difference between 18 and 9: $18 - 9 = 9$
• Difference between 27 and 18: $27 - 18 = 9$
• Difference between 36 and 27: $36 - 27 = 9$
• Difference between 45 and 36: $45 - 36 = 9$
• Difference between 54 and 45: $54 - 45 = 9$

The second differences are constant and equal to 9. This means the first differences form an arithmetic progression with a common difference of 9.

Predicting the Next Difference and Next Number

Since the second difference is always 9, the next difference in the first difference sequence must be $54 + 9 = 63$.

The next number in the original series is found by adding this next difference to the last term of the original series (192).

Next number = Last term + Next difference

Next number = $192 + 63$

Next number = $255$

Thus, the number that replaces the question mark (?) is 255.

Summary of the Pattern and Solution

The pattern in the number series is that the difference between consecutive terms increases by 9 each time. We can summarize this in a table:

The next number in the series is 255.

Revision Table: Key Concepts for Number Series

Additional Information on Solving Number Patterns

Solving number series problems often involves looking for different types of patterns. While the difference method worked here, other common patterns include:

• Arithmetic Progression: A constant difference between terms (e.g., 2, 5, 8, 11...).
• Geometric Progression: A constant ratio between terms (e.g., 3, 6, 12, 24...).
• Squares or Cubes: Terms might be squares ($1, 4, 9, 16...$) or cubes ($1, 8, 27, 64...$) or related to them.
• Combinations: Patterns might involve a combination of addition/subtraction and multiplication/division, or alternating operations.
• Fibonacci Series: Each term is the sum of the two preceding terms (e.g., 1, 1, 2, 3, 5, 8...).

Sometimes, the pattern is in the differences, as seen in this problem, or even in the differences of the differences (second differences, third differences, etc.). Always start by calculating the differences to see if a simple pattern emerges.