Number Series Pattern Explained

The question asks us to find the missing number in the series: 7, 11, 14, 25, 29, 55, ?

To solve this number series question, we need to identify the pattern or rule that governs the sequence of numbers.

Analyzing the Given Series

Let the terms of the series be denoted by \(a_n\), where \(n\) is the position of the term in the series. The given series is:

• \(a_1 = 7\)
• \(a_2 = 11\)
• \(a_3 = 14\)
• \(a_4 = 25\)
• \(a_5 = 29\)
• \(a_6 = 55\)
• \(a_7 = ?\)

Discovering the Pattern

Let's examine the relationships between consecutive and non-consecutive terms to find the underlying number series pattern.

We can observe the following relationships:

• The fourth term (\(a_4\)) seems related to the sum of the second and third terms: \(a_2 + a_3 = 11 + 14 = 25\). This matches \(a_4\).
• The fifth term (\(a_5\)) might be related to the third term (\(a_3\)): \(a_3 \times 2 + 1 = 14 \times 2 + 1 = 28 + 1 = 29\). This matches \(a_5\).
• The sixth term (\(a_6\)) might be related to the fourth term (\(a_4\)): \(a_4 \times 2 + 5 = 25 \times 2 + 5 = 50 + 5 = 55\). This matches \(a_6\).

Based on these observations, we can propose a pattern where the rule for finding the next term depends on its position:

• For \(n=4\), \(a_n = a_{n-2} + a_{n-1}\)
• For \(n \ge 5\) and \(n\) is odd, \(a_n = a_{n-2} \times 2 + 1\)
• For \(n \ge 6\) and \(n\) is even, \(a_n = a_{n-2} \times 2 + 5\)

Verifying the Pattern

Let's check if this pattern holds for the given terms:

• \(a_1 = 7\) (Given)
• \(a_2 = 11\) (Given)
• \(a_3 = 14\) (Given)
• For \(n=4\): \(a_4 = a_{4-2} + a_{4-1} = a_2 + a_3 = 11 + 14 = 25\). This matches the given \(a_4\).
• For \(n=5\) (odd, \(\ge 5\)): \(a_5 = a_{5-2} \times 2 + 1 = a_3 \times 2 + 1 = 14 \times 2 + 1 = 28 + 1 = 29\). This matches the given \(a_5\).
• For \(n=6\) (even, \(\ge 6\)): \(a_6 = a_{6-2} \times 2 + 5 = a_4 \times 2 + 5 = 25 \times 2 + 5 = 50 + 5 = 55\). This matches the given \(a_6\).

The pattern successfully generates the given terms in the series.

Calculating the Missing Term

We need to find the term at position \(n=7\). Since \(n=7\) is odd and \(\ge 5\), we use the rule for odd indices \(\ge 5\):

\(a_7 = a_{7-2} \times 2 + 1 = a_5 \times 2 + 1\)

We know that \(a_5 = 29\). Substituting this value into the formula:

\(a_7 = 29 \times 2 + 1\)

\(a_7 = 58 + 1\)

\(a_7 = 59\)

Therefore, the missing number in the series is 59.

Revision Table: Number Series Pattern

Additional Information: Solving Number Series Questions

Number series questions are common in logical reasoning and quantitative aptitude tests. They require you to find the underlying pattern in a sequence of numbers.

Common types of patterns include:

• Arithmetic progression (constant difference)
• Geometric progression (constant ratio)
• Difference series (differences between terms form a pattern)
• Ratio series (ratios between terms form a pattern)
• Mixed series (combination of arithmetic and geometric operations)
• Fibonacci-like series (terms are sums of previous terms)
• Alternating series (different patterns for alternate terms)
• Patterns based on squares, cubes, prime numbers, etc.

Solving strategy often involves calculating differences between terms, looking for ratios, checking for alternating patterns, or trying combinations of operations. It's important to be systematic and try different possibilities until a consistent rule is found for the entire given sequence.