Finding the Missing Number in the Series

Let's analyze the given number series to find the pattern and determine the missing number: 1, 3, 17, 55, ?, 179, 265, 375.

To identify the pattern in a number series, we often look at the differences between consecutive terms. Let's calculate the first differences:

• Difference between 3 and 1: $3 - 1 = 2$
• Difference between 17 and 3: $17 - 3 = 14$
• Difference between 55 and 17: $55 - 17 = 38$
• Difference between 179 and 265: $265 - 179 = 86$
• Difference between 265 and 375: $375 - 265 = 110$

So the series of first differences is: 2, 14, 38, ?, ?, 86, 110.

Since the first differences do not show a simple arithmetic progression, let's calculate the differences between these first differences (the second differences):

• Difference between 14 and 2: $14 - 2 = 12$
• Difference between 38 and 14: $38 - 14 = 24$
• Difference between 110 and 86: $110 - 86 = 24$

The second differences we have calculated are 12, 24, and 24. Let's arrange the differences in a table to see the pattern clearly:

Looking at the known second differences (12, 24, ?, ?, ?, 24), a pattern emerges: 12, 24, 12, 24, 12, 24... This seems to be an alternating pattern of 12 and 24.

Following this pattern for the second differences:

• The third second difference should be 12. So, $d_4 - 38 = 12$. This means $d_4 = 38 + 12 = 50$.
• The fourth second difference should be 24. So, $d_5 - d_4 = 24$. This means $d_5 - 50 = 24$, so $d_5 = 50 + 24 = 74$.
• Let's check the next second difference: $86 - d_5$. If $d_5 = 74$, then $86 - 74 = 12$. This matches the expected pattern (12, 24, 12, 24, 12, 24...).

So, the series of first differences is 2, 14, 38, 50, 74, 86, 110.

Now we can find the missing term in the original series using the first differences:

• The missing term is the term after 55. The difference between the missing term and 55 is $d_4$, which is 50.
• Missing term = $55 + d_4 = 55 + 50 = 105$.

Let's verify this with the next term. The difference between 179 and the missing term should be $d_5$, which is 74.

• $179 - 105 = 74$. This is correct.

Thus, the missing number in the series is 105.

The complete series with the missing number is: 1, 3, 17, 55, 105, 179, 265, 375.

Number Series Analysis and Pattern Identification

Let's summarise the series and the differences found:

The pattern in the second differences (12, 24, 12, 24, 12, 24) confirms our answer.

Revision Table: Number Series Solution

Additional Information: Types of Number Series Patterns

Number series questions in logical reasoning can follow various patterns. Understanding common types helps in solving them:

• Arithmetic Series: Constant difference between terms (constant first difference). Example: 2, 5, 8, 11... (difference is 3).
• Geometric Series: Constant ratio between terms (each term is multiplied by a fixed number to get the next). Example: 3, 6, 12, 24... (ratio is 2).
• Differences of Differences: The pattern emerges only after calculating first, second, or higher-order differences, like in this question.
• Squares/Cubes/Powers: Terms might be squares, cubes, or other powers of numbers, or related to them (e.g., $n^2$, $n^2 \pm k$, $n^3$, $n^3 \pm k$). Example: 4, 9, 16, 25... ($2^2, 3^2, 4^2, 5^2$).
• Mixed Operations: The pattern might involve a combination of operations (e.g., multiply by a number and then add/subtract another number, or operations based on the position of the term). Example: 2, 5, 11, 23... (Multiply by 2 and add 1: $2\times2+1=5$, $5\times2+1=11$, $11\times2+1=23$).
• Fibonacci or Similar Series: Each term is the sum of the previous two terms (or other specific combinations of previous terms). Example: 1, 1, 2, 3, 5, 8... ($1+1=2$, $1+2=3$, $2+3=5$).
• Alternating Series: The pattern might alternate between two different rules or sequences.

Solving number series problems often involves trying out different approaches, starting with calculating differences, until a consistent pattern is found.