Analyzing the Given Number Series Question

The question asks us to find the number that replaces the question mark (?) in the given series: 18, 34, 66, 130, ?. To solve this number series question, we need to identify the underlying pattern or rule that connects the terms.

Identifying the Pattern in the Number Series

Let's look at the relationship between consecutive terms in the series 18, 34, 66, 130, ?.

We can examine the differences between the terms:

• Difference between the 2nd and 1st term: $34 - 18 = 16$
• Difference between the 3rd and 2nd term: $66 - 34 = 32$
• Difference between the 4th and 3rd term: $130 - 66 = 64$

The differences between consecutive terms are 16, 32, and 64. We can observe a pattern in these differences:

• $16 \times 2 = 32$
• $32 \times 2 = 64$

The pattern in the differences is that each successive difference is double the previous one. Following this pattern, the next difference should be $64 \times 2 = 128$.

Alternatively, let's look for a direct relationship between a term and the next term:

• $18 \times 2 - 2 = 36 - 2 = 34$
• $34 \times 2 - 2 = 68 - 2 = 66$
• $66 \times 2 - 2 = 132 - 2 = 130$

This pattern also holds true: each term is obtained by multiplying the previous term by 2 and then subtracting 2. This is a consistent pattern observed across the given numbers in the series.

Step-by-Step Solution to Find the Missing Number

Using the identified pattern (each term is 2 times the previous term minus 2), we can find the next number in the series after 130.

The last term given is 130. Applying the rule:

Next term = (Last term $\times$ 2) - 2

Next term = ($130 \times 2$) - 2

Next term = $260 - 2$

Next term = $258$

Using the pattern of differences:

The differences are 16, 32, 64. The next difference is $64 \times 2 = 128$.

The next term is the last term plus the next difference:

Next term = $130 + 128 = 258$.

Both identified patterns lead to the same result, 258, as the number that replaces the question mark in the series.

Conclusion: The Number to Replace the Question Mark

Based on the pattern analysis of the given number series 18, 34, 66, 130, ?, the number that replaces the question mark is 258.

Revision Table: Key Concepts in Number Series

Additional Information: Solving Number Series Problems

Solving number series questions is a common part of reasoning and quantitative aptitude tests. Here are some tips for approaching these problems:

• Calculate the differences between consecutive terms. Look for a pattern in these differences (arithmetic progression, geometric progression, squares, cubes, etc.).
• Calculate the ratios between consecutive terms. See if there's a consistent multiplier or divisor.
• Look for alternating patterns (e.g., add then subtract, multiply then divide).
• Consider patterns involving squares, cubes, square roots, or cube roots.
• Sometimes, the pattern involves operations on the term number itself (e.g., $n^2 + 1$).
• Practice with various types of series to become familiar with common patterns.
• If one approach doesn't reveal a clear pattern, try another. Don't get stuck on just looking at differences or ratios.

The key is systematic observation and testing potential patterns based on the given numbers in the series.