Analyzing the Number Series Pattern

The question asks us to find the missing number in the given series: 15, ?, 27, 39, 55, 75. To solve number series questions, we often look for a pattern in the differences or ratios between consecutive terms.

Let's examine the differences between the known consecutive terms:

• Difference between 39 and 27: $39 - 27 = 12$
• Difference between 55 and 39: $55 - 39 = 16$
• Difference between 75 and 55: $75 - 55 = 20$

The differences we found are 12, 16, and 20. Let's look at the differences between these differences:

• Difference between 16 and 12: $16 - 12 = 4$
• Difference between 20 and 16: $20 - 16 = 4$

We observe a consistent pattern in the differences: the difference between consecutive terms increases by 4 each time. This is a series where the differences between consecutive terms form an arithmetic progression (4, 8, 12, 16, 20, ...).

Finding the Missing Number in the Series

Following this pattern, the difference between the second term (?) and the third term (27) should be the number just before 12 in the sequence of differences, which is $12 - 4 = 8$.

So, $27 - ? = 8$.

To find the missing number (?), we can rearrange the equation:

$? = 27 - 8 = 19$

Let's verify this by checking the difference between the first term (15) and the second term (19). The difference should be the number just before 8 in the sequence of differences, which is $8 - 4 = 4$.

$19 - 15 = 4$

This matches the pattern of differences increasing by 4.

Verifying the Complete Number Series

The completed series is 15, 19, 27, 39, 55, 75.

Let's check the differences between consecutive terms:

• $19 - 15 = 4$
• $27 - 19 = 8$
• $39 - 27 = 12$
• $55 - 39 = 16$
• $75 - 55 = 20$

The differences are 4, 8, 12, 16, 20, which confirms our pattern of differences increasing by 4.

Here is a table summarizing the series and the differences:

The pattern of differences (4, 8, 12, 16, 20) confirms that the missing number is 19.

Revision Table: Key Concepts for Number Series

Additional Information: Types of Number Series Patterns

Number series problems can follow various patterns. Understanding these types helps in solving different questions:

• Arithmetic Series: The difference between consecutive terms is constant. Example: 2, 5, 8, 11, ... (common difference is 3).
• Geometric Series: Each term is found by multiplying the previous term by a constant ratio. Example: 3, 6, 12, 24, ... (common ratio is 2).
• Difference Series: The differences between consecutive terms form a separate pattern (like the one in this question, where differences form an arithmetic series).
• Double Difference Series: The differences between the differences have a pattern. (Our question is an example where the second differences are constant).
• Fibonacci Series: Each term is the sum of the two preceding terms. Example: 0, 1, 1, 2, 3, 5, 8, ...
• Mixed Series: A combination of two or more patterns within a single series.
• Square/Cube Series: Terms are squares or cubes of natural numbers, or related to them. Example: 1, 4, 9, 16, ... ($1^2, 2^2, 3^2, 4^2, ...$)

Solving number series problems requires careful observation and testing different potential patterns.