Finding the Odd One Out in the Number Series

The question asks us to identify the number that does not fit the pattern in the given series: 6, 24, 60, 120, 211, 336.

To solve this, we need to analyze the series and find the underlying mathematical pattern or rule that generates the terms.

Analyzing the Number Series Pattern

Let's look for relationships between the terms. Common patterns involve arithmetic progression, geometric progression, differences, products, squares, cubes, or combinations of these.

Consider the differences between consecutive terms:

• $24 - 6 = 18$
• $60 - 24 = 36$
• $120 - 60 = 60$
• $211 - 120 = 91$
• $336 - 211 = 125$

The differences are 18, 36, 60, 91, 125. Let's look at the differences between these differences (second differences):

• $36 - 18 = 18$
• $60 - 36 = 24$
• $91 - 60 = 31$
• $125 - 91 = 34$

The second differences are 18, 24, 31, 34. These are not constant or in a simple progression, suggesting the pattern might not be based on constant differences.

Let's consider patterns related to cubes or products of consecutive numbers.

Let's examine if the terms are related to products of consecutive integers:

• $6 = 1 \times 2 \times 3$
• $24 = 2 \times 3 \times 4$
• $60 = 3 \times 4 \times 5$
• $120 = 4 \times 5 \times 6$
• What should be the next term based on this pattern? $5 \times 6 \times 7 = 210$. The given term is 211.
• What about the term after that? $6 \times 7 \times 8 = 336$. The given term is 336.

This pattern seems to fit most of the numbers. Let's define the pattern as the product of three consecutive integers, starting with $n$ for the $n$-th term, where $n$ begins from 1. The general term $T_n$ could be $T_n = n(n+1)(n+2)$.

Applying the Pattern to the Series

Using the pattern $T_n = n(n+1)(n+2)$, let's generate the expected terms:

• For $n=1$: $T_1 = 1 \times (1+1) \times (1+2) = 1 \times 2 \times 3 = 6$ (Matches the first term)
• For $n=2$: $T_2 = 2 \times (2+1) \times (2+2) = 2 \times 3 \times 4 = 24$ (Matches the second term)
• For $n=3$: $T_3 = 3 \times (3+1) \times (3+2) = 3 \times 4 \times 5 = 60$ (Matches the third term)
• For $n=4$: $T_4 = 4 \times (4+1) \times (4+2) = 4 \times 5 \times 6 = 120$ (Matches the fourth term)
• For $n=5$: $T_5 = 5 \times (5+1) \times (5+2) = 5 \times 6 \times 7 = 210$ (Expected term is 210)
• For $n=6$: $T_6 = 6 \times (6+1) \times (6+2) = 6 \times 7 \times 8 = 336$ (Matches the sixth term)

Let's compare the given series with the terms generated by the pattern $T_n = n(n+1)(n+2)$:

All terms follow the pattern $n(n+1)(n+2)$ except for the fifth term. According to the pattern, the fifth term should be 210, but it is given as 211.

Identifying the Odd Number

Based on the analysis, the number 211 does not fit the established pattern of the series. The other numbers (6, 24, 60, 120, 336) are generated by the rule $T_n = n(n+1)(n+2)$ for $n=1, 2, 3, 4, 6$ respectively (or $n=1, 2, 3, 4, 5, 6$ if 211 were 210). Since 211 is the only number that deviates from this pattern, it is the odd one out.

Conclusion

The number that is the odd one out in the series 6, 24, 60, 120, 211, 336 is 211.

Revision Table: Number Series Analysis

Additional Information on Number Series Patterns

Solving number series problems often involves identifying the relationship between consecutive terms or finding a formula for the n-th term. Some common types of patterns include:

• Arithmetic Progression: Constant difference between terms.
• Geometric Progression: Constant ratio between terms.
• Differences: Patterns in the differences between consecutive terms (first difference, second difference, etc.).
• Products/Divisions: Terms might be products or divisions of previous terms or related to term number.
• Squares and Cubes: Terms related to squares or cubes of natural numbers, possibly with additions or subtractions (e.g., $n^2+c$, $n^3-n$).
• Combinations: A mix of arithmetic and geometric operations.
• Alternating Patterns: Different patterns applied to alternate terms.

Practice with various types of series is key to quickly recognizing patterns during exams or quizzes.