Which number does not belong in the series below?
64
Series Pattern Analysis: Identifying the Outlier
The question asks us to find the number that does not fit the pattern in the given number series: 2, 5, 10, 17, 26, 37, 50, 64. Let's examine the relationship between the position of a number in the series and the number itself.
Finding the Underlying Mathematical Rule
We can test a common pattern for number series: relating the term number (n) to the term value. Let's see if the terms follow the rule $n^2 + 1$.
- For the 1st term (n=1): $1^2 + 1 = 1 + 1 = 2$. This matches the first number in the series.
- For the 2nd term (n=2): $2^2 + 1 = 4 + 1 = 5$. This matches the second number.
- For the 3rd term (n=3): $3^2 + 1 = 9 + 1 = 10$. This matches the third number.
- For the 4th term (n=4): $4^2 + 1 = 16 + 1 = 17$. This matches the fourth number.
- For the 5th term (n=5): $5^2 + 1 = 25 + 1 = 26$. This matches the fifth number.
- For the 6th term (n=6): $6^2 + 1 = 36 + 1 = 37$. This matches the sixth number.
- For the 7th term (n=7): $7^2 + 1 = 49 + 1 = 50$. This matches the seventh number.
Checking the Final Number
Now let's check the eighth number in the series using the same pattern:
- For the 8th term (n=8): According to the pattern $n^2 + 1$, the number should be $8^2 + 1 = 64 + 1 = 65$.
The series provided has 64 as the eighth number, but the established pattern ($n^2 + 1$) predicts 65 for the eighth position. Therefore, 64 is the number that does not belong in this series.
Step-by-Step Calculation Summary
Here's a summary of the terms versus the pattern $n^2 + 1$:
| Position (n) | Series Number | Pattern ($n^2 + 1$) | Match? |
|---|---|---|---|
| 1 | 2 | $1^2 + 1 = 2$ | Yes |
| 2 | 5 | $2^2 + 1 = 5$ | Yes |
| 3 | 10 | $3^2 + 1 = 10$ | Yes |
| 4 | 17 | $4^2 + 1 = 17$ | Yes |
| 5 | 26 | $5^2 + 1 = 26$ | Yes |
| 6 | 37 | $6^2 + 1 = 37$ | Yes |
| 7 | 50 | $7^2 + 1 = 50$ | Yes |
| 8 | 64 | $8^2 + 1 = 65$ | No |
The analysis clearly shows that 64 deviates from the established pattern.
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