Identifying the Series Pattern

To find the missing term in the sequence 5, 17, 37, 65, ___, 145, we need to identify the mathematical rule that generates the numbers.

The pattern in this series can be described by the algebraic formula $n^2 + 1$, where 'n' is an increasing even number starting from 2.

Applying the Pattern

Let's test this formula for the given terms:

• For n = 2: $2^2 + 1 = 4 + 1 = 5$
• For n = 4: $4^2 + 1 = 16 + 1 = 17$
• For n = 6: $6^2 + 1 = 36 + 1 = 37$
• For n = 8: $8^2 + 1 = 64 + 1 = 65$

Finding the Missing Term

The pattern holds, with 'n' increasing by 2 for each term. The next term in the sequence of n (2, 4, 6, 8, ...) is 10.

For n = 10: $10^2 + 1 = 100 + 1 = 101$

To verify this, the final term in the series should correspond to the next even number, n = 12:

For n = 12: $12^2 + 1 = 144 + 1 = 145$. This matches the last term.

Conclusion

The missing number in the series is 101.