Understanding the Missing Number Series Pattern

The question asks us to find the missing number in the given series: 13, 24, 37, 61, 98, ?

To solve this missing number problem, we need to identify the underlying pattern or rule that governs the sequence of numbers. Let's examine the relationship between consecutive terms in the series.

Analyzing the Number Series

Let the terms of the series be \(T_1, T_2, T_3, T_4, T_5, T_6\).

• \(T_1 = 13\)
• \(T_2 = 24\)
• \(T_3 = 37\)
• \(T_4 = 61\)
• \(T_5 = 98\)
• \(T_6 = ?\) (the missing number)

Let's look at the differences between consecutive terms:

• \(T_2 - T_1 = 24 - 13 = 11\)
• \(T_3 - T_2 = 37 - 24 = 13\)
• \(T_4 - T_3 = 61 - 37 = 24\)
• \(T_5 - T_4 = 98 - 61 = 37\)

The differences are 11, 13, 24, 37. This sequence of differences (11, 13, 24, 37) itself does not show a simple arithmetic or geometric progression pattern easily.

Identifying the Fibonacci-like Series Pattern

Let's look for another type of pattern, such as a relationship involving the sum of previous terms.

• Consider the sum of the first two terms: \(T_1 + T_2 = 13 + 24 = 37\). This sum is equal to the third term, \(T_3\).
• Consider the sum of the second and third terms: \(T_2 + T_3 = 24 + 37 = 61\). This sum is equal to the fourth term, \(T_4\).
• Consider the sum of the third and fourth terms: \(T_3 + T_4 = 37 + 61 = 98\). This sum is equal to the fifth term, \(T_5\).

This reveals a clear pattern: each term in the series, starting from the third term, is the sum of the two preceding terms. This type of sequence is similar to the Fibonacci sequence.

The general rule for this series can be expressed as \(T_n = T_{n-1} + T_{n-2}\) for \(n \ge 3\).

Calculating the Missing Number

Following the identified pattern, the next term (\(T_6\)), which is the missing number, should be the sum of the fifth term (\(T_5\)) and the fourth term (\(T_4\)).

• \(T_6 = T_5 + T_4\)
• \(T_6 = 98 + 61\)
• \(T_6 = 159\)

Thus, the missing number in the series is 159.

Verifying the Pattern

The series generated by this rule starts with 13, 24. Then:

• \(13 + 24 = 37\)
• \(24 + 37 = 61\)
• \(37 + 61 = 98\)
• \(61 + 98 = 159\)

The series is 13, 24, 37, 61, 98, 159. This matches the given sequence and successfully determines the missing number.

Final Answer Determination

Based on the analysis, the missing number in the series 13, 24, 37, 61, 98, ? is 159.

Revision Table: Missing Number Series

Additional Information: Types of Number Series Patterns

Number series questions in aptitude tests often follow various patterns. Recognizing these patterns is key to solving missing number problems. Some common patterns include:

• Arithmetic Progression (AP): The difference between consecutive terms is constant. Example: 2, 5, 8, 11, ... (common difference is 3)
• Geometric Progression (GP): Each term is obtained by multiplying the previous term by a constant ratio. Example: 3, 6, 12, 24, ... (common ratio is 2)
• Difference Series: The differences between consecutive terms form a pattern (e.g., an AP or GP). Example: 1, 2, 4, 7, 11, ... (differences are 1, 2, 3, 4)
• Sum of Previous Terms: As seen in this problem, each term is the sum of the preceding two or more terms (Fibonacci-like).
• Alternating Series: Two different patterns alternate within the same series. Example: 5, 10, 7, 14, 9, 18, ... (Pattern 1: +5, +7, +9...; Pattern 2: *2)
• Square/Cube Series: Terms are squares or cubes of natural numbers, sometimes with additions or subtractions. Example: 1, 4, 9, 16, ... (\(n^2\)); 2, 9, 28, 65, ... (\(n^3 + 1\))
• Mixed Series: A combination of two or more different patterns.

Practicing with various types of number series helps in quickly identifying the pattern and finding the missing number.