Finding the Pattern in the Number Series

Let's analyze the given number series: 10, 15, 25, 45, 85, ?. To find the number that replaces the question mark, we need to identify the pattern or rule that governs the sequence.

Analyzing the Differences Between Consecutive Terms

We can start by looking at the difference between each term and the one preceding it:

• Difference between the 2nd and 1st term: \(15 - 10 = 5\)
• Difference between the 3rd and 2nd term: \(25 - 15 = 10\)
• Difference between the 4th and 3rd term: \(45 - 25 = 20\)
• Difference between the 5th and 4th term: \(85 - 45 = 40\)

Identifying the Pattern in the Differences

Let's look at the sequence of differences we found: 5, 10, 20, 40.

We can observe a clear pattern in these differences. Each difference is double the previous difference:

• \(5 \times 2 = 10\)
• \(10 \times 2 = 20\)
• \(20 \times 2 = 40\)

This indicates that the differences themselves form a geometric progression with a common ratio of 2.

Calculating the Next Difference

Following the pattern, the next difference in the sequence of differences should be double the last difference (40):

\(\text{Next Difference} = 40 \times 2 = 80\)

Calculating the Next Number in the Series

To find the number that replaces the question mark, we add this next difference (80) to the last term in the original series (85):

\(\text{Next Term} = \text{Last Term} + \text{Next Difference}\)

\(\text{Next Term} = 85 + 80 = 165\)

Therefore, the number that replaces the question mark is 165.

Summary of the Series and Pattern

Revision Table: Understanding Number Series Patterns

Additional Information: Solving Logical Reasoning Series Questions

Solving number series questions requires careful observation and analysis to identify the underlying pattern. Here are some common strategies:

• Look at the differences: Calculate the difference between consecutive terms. If the differences are constant, it's an arithmetic series. If they follow a pattern (like doubling or tripling), it's a difference series.
• Look at the ratios: Calculate the ratio between consecutive terms. If the ratio is constant, it's a geometric series.
• Consider squares or cubes: See if the terms are related to squares or cubes of natural numbers or their variations (\(n^2\), \(n^2+1\), \(n^3\), \(n^3-1\), etc.).
• Check for alternating patterns: Sometimes, there are two different patterns applied to alternate terms.
• Look for patterns in consecutive differences: Sometimes, the pattern is not in the first level of differences but in the differences of the differences.
• Consider combinations: The pattern might involve more than one operation (e.g., multiply by a number and then add/subtract another number).

Practicing various types of series questions helps in quickly recognizing common patterns.