Analyzing the Number Series Pattern

The question asks us to find the missing number in the sequence: 55, 58, 64, ?, 85.

To solve a number series problem, we need to identify the underlying pattern connecting the terms. Let's examine the differences between consecutive terms in the given series.

Finding the Differences Between Terms

• Difference between the 2nd term (58) and the 1st term (55): \(58 - 55 = 3\)
• Difference between the 3rd term (64) and the 2nd term (58): \(64 - 58 = 6\)

The sequence of differences starts with 3, then 6. This suggests that the differences might be increasing.

Identifying the Pattern in Differences

The differences found are 3 and 6. We can see that the second difference (6) is greater than the first difference (3) by 3 (\(6 - 3 = 3\)).

Let's hypothesize that the difference between consecutive terms increases by 3 each time.

• The first difference is 3.
• The second difference is \(3 + 3 = 6\).
• Following this pattern, the third difference should be \(6 + 3 = 9\).
• The fourth difference should be \(9 + 3 = 12\).

So, the expected sequence of differences is 3, 6, 9, 12, and so on.

Calculating the Missing Number

Using the identified pattern of differences (3, 6, 9, 12...):

• Term 1: 55
• Term 2: Term 1 + 3 = \(55 + 3 = 58\) (Matches the given series)
• Term 3: Term 2 + 6 = \(58 + 6 = 64\) (Matches the given series)
• Term 4 (the missing number): Term 3 + 9 = \(64 + 9 = 73\)
• Term 5: Term 4 + 12 = \(73 + 12 = 85\) (Matches the given series)

The pattern holds true for all given terms in the number series. Therefore, the missing number is 73.

Verification of the Number Series

Let's write down the series with the calculated number and the differences:

55 \(\xrightarrow{+3}\) 58 \(\xrightarrow{+6}\) 64 \(\xrightarrow{+9}\) 73 \(\xrightarrow{+12}\) 85

The differences are indeed 3, 6, 9, and 12, which follow the pattern where each subsequent difference increases by 3.

Conclusion on Finding the Missing Term

Based on the detailed analysis of the pattern of differences in the number series, the number that replaces the question mark (?) is 73.

Revision Table: Number Series Analysis

Additional Information on Number Series Patterns

Number series questions are common in aptitude and reasoning tests. They involve finding a pattern in a sequence of numbers. Common patterns include:

• Arithmetic Progression: The difference between consecutive terms is constant (e.g., 2, 4, 6, 8... difference is 2).
• Geometric Progression: Each term is multiplied by a constant ratio to get the next term (e.g., 3, 6, 12, 24... ratio is 2).
• Difference Series: The differences between consecutive terms form their own pattern (as seen in this question), which could be an arithmetic progression, geometric progression, squares, cubes, etc.
• Ratio Series: The ratio between consecutive terms forms a pattern.
• Mixed Series: A combination of different patterns or two interleaved series.
• Square/Cube Series: Terms are squares or cubes, or based on squares/cubes (\(n^2\), \(n^2+1\), \(n^3\), \(n^3-1\), etc.).
• Fibonacci Series: Each term is the sum of the two preceding terms (e.g., 1, 1, 2, 3, 5, 8...).

Solving number series problems often involves calculating differences, ratios, or looking for relationships between terms, squares, cubes, or prime numbers.