Let's analyze the given number series to find the pattern and determine the missing number. The series is: 82, 79, 71, ?, 60, 57, 49.

Analyzing the Number Series Pattern

To find the rule governing this series, we can look at the differences between consecutive terms:

• The difference between the first and second term is $82 - 79 = 3$.
• The difference between the second and third term is $79 - 71 = 8$.
• Let's look at the later terms where the numbers are known.
• The difference between the fifth and sixth term is $60 - 57 = 3$.
• The difference between the sixth and seventh term is $57 - 49 = 8$.

From these calculations, we can see a repeating pattern in the differences between consecutive terms: the series alternates between subtracting 3 and subtracting 8.

Finding the Missing Term in the Series

Following the identified pattern (subtract 3, then subtract 8, repeat), the difference between the third term (71) and the missing term (?) should be 3, because the previous difference was 8.

• The pattern is: $82 \xrightarrow{-3} 79 \xrightarrow{-8} 71 \xrightarrow{-3} ? \xrightarrow{-8} 60 \xrightarrow{-3} 57 \xrightarrow{-8} 49$.
• So, to find the missing number, we subtract 3 from 71:
• Missing number $= 71 - 3 = 68$.

Verifying the Number Series Pattern

Now, let's check if placing 68 in the series maintains the pattern for the subsequent terms.

• If the missing number is 68, the next difference in the pattern should be 8 (since the previous difference was 3).
• Let's check the difference between 68 and the next term, 60: $68 - 60 = 8$.
• This matches the expected pattern of subtracting 8.

The pattern successfully holds throughout the series with 68 as the missing number.

Conclusion: The Missing Number

Based on the pattern of alternating subtractions of 3 and 8, the number that replaces the question mark (?) in the series 82, 79, 71, ?, 60, 57, 49 is 68.

Revision Table: Number Series Analysis

Additional Information on Number Series

Number series questions are common in various tests and exams. They test your ability to identify patterns and rules. Common types of patterns include:

• Arithmetic Series: A constant difference between consecutive terms (e.g., 2, 4, 6, 8...).
• Geometric Series: A constant ratio between consecutive terms (e.g., 3, 9, 27, 81...).
• Difference Series: The differences between consecutive terms follow a pattern (which could be another series). This is the type we saw in this problem.
• Alternating Series: The pattern involves operations that alternate between terms, or the series might interleave two different patterns.
• Fibonacci-like Series: Each term is the sum of the previous two terms (e.g., 1, 1, 2, 3, 5, 8...).

Solving these requires careful observation and often involves calculating differences, ratios, or looking for other relationships between terms.