Finding the Missing Number in a Series

This question asks us to identify the number that should replace the question mark (?) in the given number series: 39, 42, 48, 57, 69, ?, 102. To solve this type of number series problem, we need to find the underlying pattern connecting the consecutive terms.

Analyzing the Differences Between Terms

Let's calculate the difference between each consecutive pair of numbers in the series:

• Difference between the 2nd and 1st term: 42 - 39 = 3
• Difference between the 3rd and 2nd term: 48 - 42 = 6
• Difference between the 4th and 3rd term: 57 - 48 = 9
• Difference between the 5th and 4th term: 69 - 57 = 12

Let the missing term be represented by x. The next differences would be:

• Difference between the missing term and the 5th term: x - 69
• Difference between the 7th and the missing term: 102 - x

Identifying the Pattern in the Differences

Let's look at the sequence of differences we calculated: 3, 6, 9, 12. This sequence itself appears to follow a pattern. The differences are increasing by a constant value.

• Difference between the 2nd and 1st difference: 6 - 3 = 3
• Difference between the 3rd and 2nd difference: 9 - 6 = 3
• Difference between the 4th and 3rd difference: 12 - 9 = 3

It's clear that the difference between consecutive terms is increasing by 3 each time. This is a common type of pattern called a second-order arithmetic series.

Predicting the Next Differences

Following this pattern, the next difference in the sequence of differences (3, 6, 9, 12, ...) should be 12 + 3 = 15.

This means the difference between the 5th term (69) and the missing term (?) should be 15.

So, x - 69 = 15.

Calculating the Missing Term

To find the missing term (x), we can add the predicted difference (15) to the last known term (69):

x = 69 + 15

x = 84

Verifying the Pattern with the Last Term

If the missing term is 84, the next difference in the main series should be between 102 and 84. The next difference in the difference sequence (3, 6, 9, 12, 15, ...) should be 15 + 3 = 18.

Let's check if 102 - 84 equals 18.

102 - 84 = 18

This confirms that our calculated missing term (84) fits the established pattern in the series.

The completed number series is: 39, 42, 48, 57, 69, 84, 102.

Summary of the Series Pattern

The differences between consecutive terms are 3, 6, 9, 12, 15, 18, which is an arithmetic progression with a common difference of 3.

Therefore, the number that replaces the question mark is 84.

Revision Table: Types of Number Series

Additional Information: Strategies for Solving Number Series

When faced with a number series question, try these strategies to find the pattern:

• Calculate Differences: Find the difference between consecutive terms. If the differences are constant, it's an arithmetic series. If the differences form a new, recognizable pattern (like an arithmetic series or geometric series), it's a higher-order series.
• Calculate Ratios: Divide each term by the preceding term. If the ratio is constant, it's a geometric series.
• Look for Alternating Patterns: Sometimes, the pattern alternates between two different operations or sequences.
• Check for Squares or Cubes: Terms might be related to squares (n^2) or cubes (n^3), or these plus/minus a constant.
• Combine Operations: The pattern might involve a combination of operations (e.g., multiply by 2, then add 1).
• Positional Patterns: The value of a term might depend on its position in the series (e.g., the n-th term is 2n + 1).

Systematically applying these methods will help you uncover the logic behind most number series problems.