Finding the Missing Number in a Series

Let's analyze the given number series to find the pattern and determine the missing number. The series is:

35, 50, 63, 82, ?, 122

We need to identify the rule that transforms one number into the next in the sequence.

Analyzing the Pattern in the Number Series

First, let's look at the differences between consecutive terms:

• Difference between 50 and 35: \(50 - 35 = 15\)
• Difference between 63 and 50: \(63 - 50 = 13\)
• Difference between 82 and 63: \(82 - 63 = 19\)

The differences are 15, 13, 19. This sequence of differences (15, 13, 19) doesn't immediately show a simple arithmetic or geometric progression. Let's look for other possible patterns.

Identifying a Square-Based Pattern

Let's consider if the numbers are related to perfect squares. Let's look at squares of integers near the given numbers:

• \(6^2 = 36\). \(36 - 1 = 35\)
• \(7^2 = 49\). \(49 + 1 = 50\)
  
• \(8^2 = 64\). \(64 - 1 = 63\)
  
• \(9^2 = 81\). \(81 + 1 = 82\)

This looks like a promising pattern! The pattern involves squaring consecutive integers starting from 6, and then alternately subtracting 1 and adding 1.

Applying the Pattern to Find the Missing Term

Following the identified pattern:

1. The first term, 35, is \(6^2 - 1\).
2. The second term, 50, is \(7^2 + 1\).
3. The third term, 63, is \(8^2 - 1\).
4. The fourth term, 82, is \(9^2 + 1\).

The next term in the series should follow the sequence of bases (6, 7, 8, 9, ...) and the alternating operation (-1, +1, -1, +1, ...).

The next base is 10.

The next operation after +1 is -1.

So, the missing term should be \(10^2 - 1\).

Calculation:

\(10^2 - 1 = 100 - 1 = 99\)

Verifying the Next Term

Let's check if the term after 99 fits the pattern to get 122. The base after 10 is 11. The operation after -1 is +1.

\(11^2 + 1 = 121 + 1 = 122\)

This matches the last number in the given series (122). Therefore, the pattern is confirmed, and the missing number is 99.

Conclusion: The Missing Number

The series follows the pattern of squaring consecutive integers starting from 6 and alternately subtracting 1 and adding 1. The missing number that fits this pattern is 99.

Number Series Pattern Revision

Number series problems often involve patterns based on:

• Arithmetic progression (constant difference).
• Geometric progression (constant ratio).
• Differences of differences (second-order differences).
• Squares or cubes of numbers, with additions or subtractions.
• Alternating patterns.
• Combinations of different operations.

Identifying the pattern requires careful observation and trying out different mathematical relationships between the numbers.

Additional Information on Number Series

Number series questions are common in aptitude tests and competitive exams to evaluate logical reasoning and numerical ability. Solving these problems helps improve pattern recognition skills. When you encounter a number series, consider these steps:

1. Look at the differences between consecutive terms.
2. Look at the ratio between consecutive terms.
3. If differences/ratios don't reveal a simple pattern, look at the differences of differences.
4. Consider squares, cubes, prime numbers, or other special number sequences.
5. Check for alternating patterns in operations or the numbers themselves.
6. Break down complex patterns into simpler components.

Practice with various types of series is key to mastering this topic.