Analyzing the Number Pattern Sequence

The question asks us to identify the missing number in the sequence: 14, 17, 22, 49, 61, ?, 189, 237, 317. To solve this number pattern problem, we need to find the underlying rule or pattern that governs the progression of numbers in this sequence.

Step-by-Step Pattern Discovery

Let's begin by examining the differences between consecutive terms in the given number sequence:

• The difference between the 2nd term (17) and the 1st term (14) is \(17 - 14 = 3\).
• The difference between the 3rd term (22) and the 2nd term (17) is \(22 - 17 = 5\).
• The difference between the 4th term (49) and the 3rd term (22) is \(49 - 22 = 27\).
• The difference between the 5th term (61) and the 4th term (49) is \(61 - 49 = 12\).
• The difference between the missing number (?) and the 5th term (61) is currently unknown. Let's call this difference \(A_5\).
• The difference between the 7th term (189) and the missing number (?) is also unknown. Let's call this difference \(A_6\).
• The difference between the 8th term (237) and the 7th term (189) is \(237 - 189 = 48\).
• The difference between the 9th term (317) and the 8th term (237) is \(317 - 237 = 80\).

The sequence of additions (differences between consecutive terms) we have so far is: 3, 5, 27, 12, \(A_5\), \(A_6\), 48, 80.

Identifying the Pattern in Additions

Let's denote the additions as \(A_1=3, A_2=5, A_3=27, A_4=12, A_5, A_6, A_7=48, A_8=80\). We need to find a consistent pattern within this sequence of additions.

Let's carefully look at the relationship between these addition values:

• Observe \(A_4 = 12\). Compare this to the first addition \(A_1 = 3\). Notice that \(12 = 3 \times 4\). This suggests a multiplication relationship.
• Now look at \(A_7 = 48\). Compare this to \(A_4 = 12\). Notice that \(48 = 12 \times 4\). This reinforces the multiplication by 4 idea.
• Consider \(A_8 = 80\). If we look at the sequence of additions, the term before \(A_7\) was \(A_6\), and the term before \(A_6\) was \(A_5\). If the pattern holds, \(A_8\) might be related to an earlier term multiplied by 4. \(80 = 20 \times 4\). This leads us to hypothesize that \(A_5\) (the missing addition) might be 20. If \(A_5=20\), then \(A_8 = A_5 \times 4 = 20 \times 4 = 80\). This matches the calculated \(A_8\).
• Following this logic, \(A_6\) should be related to \(A_3 = 27\) by multiplication by 4. \(A_6 = 27 \times 4 = 108\).

This observation suggests a pattern where, starting from the 4th addition, each addition is 4 times the addition three steps before it. We can express this pattern formally:

For any addition \(A_n\) where \(n \ge 4\), the value is given by \(A_n = A_{n-3} \times 4\).

Verifying the Addition Pattern

Let's confirm this rule using the known additions:

• \(A_1 = 3\) (Given)
• \(A_2 = 5\) (Given)
• \(A_3 = 27\) (Given)
• \(A_4 = A_{4-3} \times 4 = A_1 \times 4 = 3 \times 4 = 12\). This matches the difference calculated between 49 and 61.
• \(A_5 = A_{5-3} \times 4 = A_2 \times 4 = 5 \times 4 = 20\). This is the addition we need to find the missing number.
• \(A_6 = A_{6-3} \times 4 = A_3 \times 4 = 27 \times 4 = 108\). This is the addition after the missing number.
• \(A_7 = A_{7-3} \times 4 = A_4 \times 4 = 12 \times 4 = 48\). This matches the difference calculated between 189 and 237.
• \(A_8 = A_{8-3} \times 4 = A_5 \times 4 = 20 \times 4 = 80\). This matches the difference calculated between 237 and 317, and it confirms our hypothesis that \(A_5 = 20\).

The pattern for the additions is confirmed to be 3, 5, 27, 12, 20, 108, 48, 80.

Calculating the Missing Number

The missing number is the 6th term in the number sequence. It is found by adding the 5th addition (\(A_5\)) to the 5th term (\(N_5\)).

We know the 5th term is \(N_5 = 61\), and we have determined the 5th addition is \(A_5 = 20\).

Therefore, the missing number (\(N_6\)) = \(N_5 + A_5 = 61 + 20 = 81\).

Verifying the Full Sequence with the Missing Number

Let's construct the full sequence using the calculated missing number (81) and the addition pattern to ensure everything fits:

The calculated sequence perfectly matches the given sequence, confirming that the missing number is 81.

Revision Table: Number Pattern Concepts

Additional Information: Strategies for Number Pattern Problems

Solving number pattern problems often requires exploring various approaches. Here are some common strategies:

• Calculate Differences: Find the difference between consecutive terms. If no pattern is obvious, calculate the differences of the differences (second differences), and so on.
• Check Ratios: For geometric sequences, look for a common ratio between consecutive terms (divide a term by the previous one).
• Look for Squares or Cubes: The pattern might involve squaring or cubing the term position or a related number.
• Consider Alternating Patterns: The rule might apply to alternate terms, or the operation might alternate (e.g., +x, -y, +x, -y...).
• Analyze Digits: The pattern might involve the sum, product, or difference of the digits of the numbers in the sequence.
• Combine Operations: The pattern might be a combination of arithmetic and geometric operations, or involve multiple steps per term.
• Look for Famous Sequences: Check if the sequence resembles known sequences like Fibonacci, prime numbers, etc.
• Test Options: If multiple-choice options are given, insert each option into the sequence and see if a clear pattern emerges with that number in place.
• Break Down the Sequence: Sometimes, a sequence might be two or more interleaved sequences, each with its own pattern.

This particular problem demonstrates a pattern in the sequence of differences that depends recursively on earlier differences, highlighting the variety and complexity number patterns can exhibit.