Solving the Number Series Pattern

Let's analyze the given number series to find the pattern and determine the missing number. The series is:

20, 20, 22, ?, 40, 60, 90

To find the pattern in a number series, we often look at the differences between consecutive terms.

Step-by-Step Difference Calculation

Let's calculate the difference between each consecutive term:

1. Difference between the 1st and 2nd term: \(20 - 20 = 0\)
2. Difference between the 2nd and 3rd term: \(22 - 20 = 2\)
3. Difference between the 6th and 5th term: \(60 - 40 = 20\)
4. Difference between the 7th and 6th term: \(90 - 60 = 30\)

So the sequence of differences between consecutive terms is:

0, 2, (difference between 3rd and 4th term), (difference between 4th and 5th term), 20, 30

Analyzing the Differences Pattern

Let's look at the sequence of known differences: 0, 2, ?, ?, 20, 30.

Now, let's look at the differences between these differences (the second differences):

1. Difference between the 2nd and 1st difference: \(2 - 0 = 2\)
2. Difference between the 6th and 5th difference: \(30 - 20 = 10\)

We have the second differences starting with 2 and ending with 10, with steps in between that are currently unknown. Let's assume the second differences follow a simple arithmetic progression.

If the second differences increase consistently, observing the known values (2 and 10), a pattern like 2, 4, 6, 8, 10 seems possible.

Let's test this assumption for the second differences:

• 1st second difference: 2
• 2nd second difference: 4
• 3rd second difference: 6
• 4th second difference: 8
• 5th second difference: 10 (Matches our known value)

This sequence of second differences (2, 4, 6, 8, 10) seems to form a clear pattern.

Calculating the Missing First Differences

Let the sequence of first differences be \(d_1, d_2, d_3, d_4, d_5, d_6\).

• \(d_1 = 0\)
• \(d_2 = 2\)
• The second differences are \(d_2-d_1=2\), \(d_3-d_2\), \(d_4-d_3\), \(d_5-d_4\), \(d_6-d_5=10\).

Using our assumed second differences (2, 4, 6, 8, 10):

• \(d_3 - d_2 = 4 \implies d_3 - 2 = 4 \implies d_3 = 6\)
• \(d_4 - d_3 = 6 \implies d_4 - 6 = 6 \implies d_4 = 12\)
• \(d_5 - d_4 = 8 \implies d_5 - 12 = 8 \implies d_5 = 20\) (Matches our known value)
• \(d_6 - d_5 = 10 \implies d_6 - 20 = 10 \implies d_6 = 30\) (Matches our known value)

So the sequence of first differences is 0, 2, 6, 12, 20, 30.

Finding the Missing Number in the Series

The terms in the series are generated by adding the consecutive differences to the previous term:

• Term 1: 20
• Term 2: \(20 + d_1 = 20 + 0 = 20\)
• Term 3: \(20 + d_2 = 20 + 2 = 22\)
• Term 4 (the missing number): \(22 + d_3 = 22 + 6 = 28\)

Let's verify this by calculating the next terms using the subsequent differences:

• Term 5: \(28 + d_4 = 28 + 12 = 40\) (Matches the given series)
• Term 6: \(40 + d_5 = 40 + 20 = 60\) (Matches the given series)
• Term 7: \(60 + d_6 = 60 + 30 = 90\) (Matches the given series)

The pattern holds true. The missing number that replaces the question mark (?) is 28.

Summary of the Pattern

The second differences form an arithmetic progression: 2, 4, 6, 8, 10. This pattern correctly generates the first differences 0, 2, 6, 12, 20, 30, which in turn generate the original series 20, 20, 22, 28, 40, 60, 90.

Conclusion

Based on the pattern of second differences, the missing number in the series 20, 20, 22, ?, 40, 60, 90 is 28.

Revision Table - Number Series Pattern Analysis

Reviewing the methods used to solve number series problems is crucial for exam preparation. Here's a quick revision table.

Additional Information - Understanding Number Series Reasoning

Number series problems are a common type of logical reasoning question. They test your ability to identify patterns and rules that govern a sequence of numbers. Mastering these problems requires practice and familiarity with different types of patterns.

Key strategies include:

• Always calculate the differences between terms first. This is the most common pattern type.
• If the first differences don't show a pattern, calculate the second differences.
• Look for patterns involving squares, cubes, or simple arithmetic operations (+, -, *, /).
• Consider alternating patterns, where the rule changes for every other term.
• Don't spend too much time on one problem; if a pattern isn't obvious quickly, move on and come back later if time permits.

Regular practice with different types of number series will help you quickly recognize patterns during exams.