Identify the number that DOES NOT belong to the following series.
64
Finding the Outlier in the Number Series
The question asks us to identify the number that does not follow the pattern in the given series: 2, 9, 28, 64, 126, 217, 344, 513. To find the outlier, we need to analyze the relationship between the numbers in the sequence and determine the underlying pattern.
Analyzing the Pattern of the Series
Let's examine the numbers and see if they relate to perfect squares or cubes. Observing the numbers like 2, 9, 28, 126, etc., they seem to be slightly more than perfect cubes.
Let's consider the sequence of natural numbers ($n$) starting from 1 and calculate $n^3$ and $n^3 + 1$:
| n | $n^3$ | $n^3 + 1$ | Given Series Number | Matches? |
|---|---|---|---|---|
| 1 | $1^3 = 1$ | $1 + 1 = 2$ | 2 | Yes |
| 2 | $2^3 = 8$ | $8 + 1 = 9$ | 9 | Yes |
| 3 | $3^3 = 27$ | $27 + 1 = 28$ | 28 | Yes |
| 4 | $4^3 = 64$ | $64 + 1 = 65$ | 64 | No |
| 5 | $5^3 = 125$ | $125 + 1 = 126$ | 126 | Yes |
| 6 | $6^3 = 216$ | $216 + 1 = 217$ | 217 | Yes |
| 7 | $7^3 = 343$ | $343 + 1 = 344$ | 344 | Yes |
| 8 | $8^3 = 512$ | $512 + 1 = 513$ | 513 | Yes |
From the table, we can see that most numbers in the series follow the pattern $n^3 + 1$, where $n$ is the position of the number in the sequence (starting from $n=1$).
- The 1st number is $1^3 + 1 = 2$.
- The 2nd number is $2^3 + 1 = 9$.
- The 3rd number is $3^3 + 1 = 28$.
- The 4th number is $4^3 + 1 = 65$. However, the series has 64 at this position.
- The 5th number is $5^3 + 1 = 126$.
- The 6th number is $6^3 + 1 = 217$.
- The 7th number is $7^3 + 1 = 344$.
- The 8th number is $8^3 + 1 = 513$.
Identifying the Number That DOES NOT Belong
Based on the pattern $n^3 + 1$, the number at the 4th position should be 65 ($4^3 + 1$). However, the given series has 64 at the 4th position. All other numbers in the series fit the $n^3 + 1$ pattern.
Therefore, the number that DOES NOT belong to the series is 64.
Conclusion
The series follows the pattern $n^3 + 1$. The number 64 is an exception as it is simply $4^3$, not $4^3 + 1$. Thus, 64 is the outlier in the given number series.
Revision Table: Series Pattern Analysis
Reviewing the series 2, 9, 28, 64, 126, 217, 344, 513, the pattern $n^3 + 1$ fits all numbers except 64.
Additional Information: Number Series and Patterns
Number series questions test your ability to identify mathematical patterns. Common patterns include:
- Arithmetic Progression: Adding a constant difference.
- Geometric Progression: Multiplying by a constant ratio.
- Differences or Ratios of differences follow a pattern.
- Squares or Cubes of numbers (like $n^2$, $n^3$) or operations involving them (like $n^2+1$, $n^3-1$, $n^2+n$).
- Alternating patterns.
- Fibonacci sequence or similar recursive relations.
To solve number series problems, it's helpful to look at the differences between terms, ratios, or try relating terms to squares, cubes, or other powers of their position number.
Important Questions from Number Series
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Select the number from among the given options that can replace the question mark (?) in the following series.
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