Solving the Number Series Puzzle

The question asks us to find the missing term in the given number series: 7, 11, 19, 31, ?, 67.

To solve a number series problem, we need to identify the pattern or the rule that connects consecutive terms in the series.

Analyzing the Number Series Pattern

Let's look at the differences between successive terms in the series:

• Difference between the second term (11) and the first term (7): \(11 - 7 = 4\)
• Difference between the third term (19) and the second term (11): \(19 - 11 = 8\)
• Difference between the fourth term (31) and the third term (19): \(31 - 19 = 12\)

Let's list the differences we found:

4, 8, 12, ...

Identifying the Pattern in Differences

Now, let's examine the sequence of differences (4, 8, 12). We can see a pattern here:

• The difference between the second difference (8) and the first difference (4): \(8 - 4 = 4\)
• The difference between the third difference (12) and the second difference (8): \(12 - 8 = 4\)

It appears that the differences between the terms are increasing by 4 each time. This is an arithmetic progression of the differences.

Predicting the Next Difference in the Number Series

Following this pattern, the next difference should be 4 more than the last difference we found (12). So, the next difference is:

\(12 + 4 = 16\)

Finding the Missing Term in the Number Series

The missing term is the term that comes after 31. According to our pattern, the difference between the missing term and 31 should be 16. Therefore, the missing term is:

\(31 + 16 = 47\)

Verifying the Number Series Pattern

Let's check if this pattern holds for the subsequent term (67). If the pattern continues, the difference between the term 67 and our calculated missing term (47) should be the next value in the sequence of differences.

The sequence of differences is 4, 8, 12, 16. The next difference should be \(16 + 4 = 20\).

Let's calculate the difference between 67 and 47:

\(67 - 47 = 20\)

Since this difference is indeed 20, our identified pattern is correct, and the missing term is 47.

Conclusion: The Missing Number

The sequence with the missing term filled in is: 7, 11, 19, 31, 47, 67.

The pattern is that each term is obtained by adding a value to the previous term, where these added values form the series 4, 8, 12, 16, 20, ...

Revision Table: Number Series Analysis

Additional Information: Types of Number Series

Number series questions are common in aptitude tests. They can follow various patterns:

• Arithmetic Series: The difference between consecutive terms is constant (e.g., 2, 4, 6, 8...).
• Geometric Series: The ratio between consecutive terms is constant (e.g., 2, 4, 8, 16...).
• Difference Series: The differences between consecutive terms form a pattern (like the one in this problem, where differences form an arithmetic series).
• Double Difference Series: The differences of the differences form a pattern.
• Mixed Series: Combining two or more patterns or interleaved series.
• Fibonacci Series: Each term is the sum of the two preceding terms (e.g., 0, 1, 1, 2, 3, 5...).
• Square or Cube Series: Terms are squares or cubes of numbers, or variations involving squares/cubes.

To solve number series problems effectively, practice identifying different types of patterns, starting with basic arithmetic or geometric progressions and then looking at the differences between terms.