The question asks us to find the missing term in the given numerical series: 1, 9, 25, 49, 81, ?, 169.

To solve this type of series problem, we need to look for a pattern or rule that connects the numbers in the sequence.

Analyzing the Numerical Series Pattern

Let's examine the numbers in the series:

• The first term is 1.
• The second term is 9.
• The third term is 25.
• The fourth term is 49.
• The fifth term is 81.
• The last term is 169.

We can try to see if there's a relationship between these numbers and their position in the series, or a relationship between consecutive numbers.

Identifying the Rule in the Series

Let's consider the possibility that these numbers are squares of some sequence of numbers:

• $1 = 1 \times 1 = 1^2$
• $9 = 3 \times 3 = 3^2$
• $25 = 5 \times 5 = 5^2$
• $49 = 7 \times 7 = 7^2$
• $81 = 9 \times 9 = 9^2$
• $169 = 13 \times 13 = 13^2$

The bases of these squares are 1, 3, 5, 7, 9, and 13. Let's list these bases in order:

1, 3, 5, 7, 9, ?, 13

Looking at this sequence of bases (1, 3, 5, 7, 9, ..., 13), we can see that these are consecutive odd numbers starting from 1.

The sequence of odd numbers is 1, 3, 5, 7, 9, 11, 13, 15, and so on.

Finding the Missing Term

Following the pattern of consecutive odd numbers, the next odd number after 9 is 11. The term missing in the original series corresponds to the square of this next odd number.

So, the missing term is $11^2$.

Calculation: $11 \times 11 = 121$.

Therefore, the missing term in the series is 121.

Verification

Let's write the series again with the calculated missing term:

1, 9, 25, 49, 81, 121, 169

Checking the bases:

• $1 = 1^2$
• $9 = 3^2$
• $25 = 5^2$
• $49 = 7^2$
• $81 = 9^2$
• $121 = 11^2$
• $169 = 13^2$

The bases are 1, 3, 5, 7, 9, 11, 13, which are indeed consecutive odd numbers. The pattern holds true.

The missing term is 121.

Revision Table: Number Series Patterns

Additional Information on Series Questions

Solving number series questions often involves identifying the underlying mathematical pattern. This pattern can be:

• A constant difference (arithmetic progression).
• A constant ratio (geometric progression).
• Differences between terms forming another simple series.
• Terms being squares, cubes, or other powers of numbers following a pattern.
• Alternating patterns.
• Combinations of different operations (e.g., multiply by a number and add/subtract another).

It's helpful to practice recognizing common sequences like squares, cubes, prime numbers, or sequences based on simple arithmetic operations.