Analyzing the Number Series Pattern

The question asks us to find the missing number in the given series: 7, 14, 25, 45, 81, ?, 239.

To solve number series problems, we usually look for a pattern in the differences between consecutive terms, the ratio between terms, or a combination of operations.

Step-by-Step Solution to Find the Missing Number

Let's find the difference between consecutive terms in the series:

• Difference between 14 and 7: $14 - 7 = 7$
• Difference between 25 and 14: $25 - 14 = 11$
• Difference between 45 and 25: $45 - 25 = 20$
• Difference between 81 and 45: $81 - 45 = 36$

The series of differences is 7, 11, 20, 36. This sequence doesn't immediately show a simple arithmetic or geometric progression. Let's examine the differences between these differences (the second differences):

• Difference between 11 and 7: $11 - 7 = 4$
• Difference between 20 and 11: $20 - 11 = 9$
• Difference between 36 and 20: $36 - 20 = 16$

The second differences are 4, 9, 16. We can observe that these numbers are perfect squares:

• $4 = 2^2$
• $9 = 3^2$
• $16 = 4^2$

This suggests a pattern where the second differences are consecutive perfect squares starting from $2^2$. The next second difference should therefore be $5^2 = 25$.

Predicting the Next Terms in the Series

Following the pattern, the next difference in the original series (after 36) should be $36 + 25 = 61$.

So, the missing term (?) is found by adding this difference (61) to the last known term (81):

Missing term = $81 + 61 = 142$.

Let's verify this by finding the difference between the term after the missing one (239) and the calculated missing term (142).

• Difference between 239 and 142: $239 - 142 = 97$

The next difference in the sequence of differences should be $61 + (\text{next second difference})$. The next second difference is $6^2 = 36$. So, the next difference should be $61 + 36 = 97$. This matches our calculation ($239 - 142 = 97$).

The complete series of differences is:

7, 11, 20, 36, 61, 97

The complete series of second differences is:

4 ($2^2$), 9 ($3^2$), 16 ($4^2$), 25 ($5^2$), 36 ($6^2$)

The series with the missing number replaced is: 7, 14, 25, 45, 81, 142, 239.

Here's a table summarizing the pattern:

The pattern clearly indicates that the missing number is 142.

Revision Table: Key Concepts

Additional Information on Solving Number Series

Solving number series problems often involves checking for multiple types of patterns. Here are some common approaches:

• Arithmetic Progression: Check if the difference between consecutive terms is constant.
• Geometric Progression: Check if the ratio between consecutive terms is constant.
• Arithmetic-Geometric Series: A combination of arithmetic and geometric progressions.
• Differences: Analyze the differences between consecutive terms. If the first differences don't work, look at second differences, third differences, and so on.
• Squares and Cubes: See if terms are related to squares or cubes of natural numbers ($n^2$, $n^3$) or variations like $n^2 \pm a$, $n^3 \pm a$.
• Prime Numbers: The series might consist of prime numbers or numbers related to them.
• Alternating Patterns: The pattern might alternate between different rules for odd and even terms.
• Combinations: The pattern might involve a combination of operations, like multiply by a number and then add/subtract another number ($an + b$).

Practice with different types of series is key to quickly identifying the underlying pattern.