Finding the Missing Number in a Series

The question asks us to find the number that replaces the question mark (?) in the given series: 98, 118, 140, ?, 190, 218.

To solve this type of number series problem, we usually look for a pattern in the differences between consecutive terms or some other mathematical relationship.

Analyzing the Pattern in the Number Series

Let's calculate the difference between each adjacent pair of numbers in the given series:

• Difference between the 2nd and 1st term: $118 - 98 = 20$
• Difference between the 3rd and 2nd term: $140 - 118 = 22$
• Difference between the 6th and 5th term: $218 - 190 = 28$

So far, the differences are 20 and 22, and the last difference is 28. Let's look at these differences: 20, 22, ..., 28. There seems to be an increasing trend in the differences.

Let's assume the differences are increasing by a constant value. The difference between the first two differences is $22 - 20 = 2$. Let's hypothesize that the differences are increasing by 2 each time.

If this hypothesis is correct, the sequence of differences should be 20, 22, 24, 26, 28.

Applying the Pattern to Find the Missing Term

Based on the hypothesized pattern of differences:

• The difference between the 1st and 2nd term is 20. ($98 + 20 = 118$)
• The difference between the 2nd and 3rd term is 22. ($118 + 22 = 140$)
• The difference between the 3rd and the missing term should be 24.
• The difference between the missing term and the 5th term (190) should be 26.
• The difference between the 5th term (190) and the 6th term (218) is 28. ($190 + 28 = 218$) - Wait, this does not match our hypothesis of the difference being 26 here. Let's re-check. The difference between 190 and 218 is indeed 28, matching the last step in the hypothesized sequence of differences (20, 22, 24, 26, 28).

Let's recalculate the differences carefully:

• 118 - 98 = 20
• 140 - 118 = 22
• 218 - 190 = 28

The differences we know are 20, 22, and 28. These are the 1st, 2nd, and 5th differences in the sequence of differences. If the differences increase by 2 each time, the full sequence of differences would be 20, 22, 24, 26, 28.

Let's verify if this sequence of differences generates the original series:

• Starting with 98:
• 98 + 20 = 118 (2nd term)
• 118 + 22 = 140 (3rd term)
• 140 + 24 = 164 (This should be the 4th term, replacing ?)
• 164 + 26 = 190 (5th term)
• 190 + 28 = 218 (6th term)

The calculated series 98, 118, 140, 164, 190, 218 matches the given series perfectly when the missing term is 164.

Therefore, the number that replaces the question mark is 164.

Summary of the Pattern

The pattern is that the difference between consecutive terms increases by 2 each time.

Revision Table: Number Series Patterns

Additional Information on Number Series Problems

Number series questions are common in logical reasoning and quantitative aptitude tests. They assess your ability to identify patterns and relationships between numbers.

Tips for solving number series problems:

• Calculate the differences between consecutive terms.
• Calculate the ratio between consecutive terms.
• Check for patterns in the differences (e.g., arithmetic progression).
• Look for patterns involving squares, cubes, prime numbers, or alternating operations.
• Consider if the series is a combination of two or more independent series.

Practice with different types of number series problems helps in quickly recognizing patterns during exams.