Number Series Puzzle: Find the Missing Term

The question asks us to identify the missing number in the given number series: 5, 10, 17, 26, 37, ?, 65.

Analyzing the Number Series Pattern

Let's examine the differences between consecutive terms in the series to find the underlying pattern:

• Difference between the 2nd and 1st term: $10 - 5 = 5$
• Difference between the 3rd and 2nd term: $17 - 10 = 7$
• Difference between the 4th and 3rd term: $26 - 17 = 9$
• Difference between the 5th and 4th term: $37 - 26 = 11$

We can observe that the differences between consecutive terms are increasing by 2 each time (5, 7, 9, 11). This suggests an arithmetic progression in the differences.

Calculating the Missing Number

Following the pattern of differences, the next difference should be $11 + 2 = 13$. Therefore, the missing number is found by adding this difference to the previous term (37):

Missing Number = $37 + 13 = 50$

Let's verify this by checking the difference between the last term (65) and our calculated missing number (50). The difference should be $13 + 2 = 15$.

Check: $65 - 50 = 15$. The pattern holds true.

Alternative Pattern Verification

Another way to look at the series is by relating it to squares:

• $5 = 2^2 + 1 \quad (4 + 1)$
• $10 = 3^2 + 1 \quad (9 + 1)$
• $17 = 4^2 + 1 \quad (16 + 1)$
• $26 = 5^2 + 1 \quad (25 + 1)$
• $37 = 6^2 + 1 \quad (36 + 1)$

The pattern appears to be $n^2 + 1$, where 'n' starts from 2 and increases by 1 for each subsequent term.

Following this pattern, the missing term corresponds to $n=7$:

Missing Number = $7^2 + 1 = 49 + 1 = 50$

The next term (65) corresponds to $n=8$:

Next Term = $8^2 + 1 = 64 + 1 = 65$. This confirms the pattern.

Conclusion

Both methods confirm that the missing number in the series is 50. This corresponds to the first option provided.