Analyzing the Number Series Pattern

The question asks us to find the next number in the given series: 39, 53, 69, 87, ?. To solve this type of problem, we need to identify the underlying pattern or rule that connects the numbers in the series.

Finding the Pattern in the Series

Let's look at the differences between consecutive terms in the series. This is a common method to find patterns in numerical sequences.

• Difference between the second and first term: \(53 - 39\)
• Difference between the third and second term: \(69 - 53\)
• Difference between the fourth and third term: \(87 - 69\)

Calculating the Differences

Performing the calculations:

• \(53 - 39 = 14\)
• \(69 - 53 = 16\)
• \(87 - 69 = 18\)

The sequence of differences is 14, 16, 18.

Identifying the Pattern in the Differences

Let's examine the sequence of differences (14, 16, 18). We can see that these differences are increasing. Let's find the difference between these differences:

• Difference between the second difference and the first difference: \(16 - 14 = 2\)
• Difference between the third difference and the second difference: \(18 - 16 = 2\)

The difference between consecutive differences is a constant value of 2. This means the differences form an arithmetic progression with a common difference of 2.

Determining the Next Difference and the Next Term

Since the differences are increasing by 2 each time (14, 16, 18), the next difference in the sequence should be \(18 + 2 = 20\). To find the next term in the original series, we need to add this next difference (20) to the last term in the series (87).

Next term = Last term + Next difference

Next term = \(87 + 20\)

Next term = \(107\)

Therefore, the number that replaces the question mark is 107.

Summary of the Pattern

The pattern in the series is that each term is obtained by adding an increasing difference to the previous term. The differences themselves form an arithmetic progression: +14, +16, +18, +20, and so on.

Revision Table: Number Series Analysis

Additional Information: Types of Number Series Patterns

Number series questions can involve various patterns. Understanding common types can help in solving them.

• Arithmetic Series: Each term is obtained by adding a constant value (common difference) to the previous term. Example: 2, 5, 8, 11, ... (common difference is 3).
• Geometric Series: Each term is obtained by multiplying the previous term by a constant value (common ratio). Example: 3, 6, 12, 24, ... (common ratio is 2).
• Series with Increasing/Decreasing Differences: The differences between consecutive terms follow their own pattern (like an arithmetic progression of differences, as seen in this problem).
• Alternating Series: The pattern alternates between two different operations or rules. Example: 1, 5, 2, 6, 3, 7, ... (+4, -3, +4, -3, ...).
• Fibonacci Series: Each term is the sum of the two preceding terms. Example: 1, 1, 2, 3, 5, 8, ... (starting from the third term).
• Prime Number Series: The terms are consecutive prime numbers. Example: 2, 3, 5, 7, 11, ...
• Square or Cube Series: The terms are squares or cubes of consecutive numbers, or variations involving squares/cubes. Example: 1, 4, 9, 16, ... (\(1^2, 2^2, 3^2, 4^2\)).

Solving number series problems often requires calculating differences, ratios, or looking for relationships between terms and their positions in the sequence.