Finding the Missing Number in a Series

This question asks us to identify the pattern in the given number series and use it to find the missing term. The series is: 12, 25, 52, 107, 218, ?

Analyzing the Number Series Pattern

Let's examine the relationship between consecutive terms in the series to find the underlying pattern. We can look at the difference between terms or try multiplication and addition/subtraction.

Let's try the multiplication and addition/subtraction approach:

1. From 12 to 25: \(12 \times 2 = 24\). \(24 + 1 = 25\).
2. From 25 to 52: \(25 \times 2 = 50\). \(50 + 2 = 52\).
3. From 52 to 107: \(52 \times 2 = 104\). \(104 + 3 = 107\).
4. From 107 to 218: \(107 \times 2 = 214\). \(214 + 4 = 218\).

The pattern seems to be: multiply the previous term by 2 and then add an increasing sequence of numbers starting from 1 (i.e., +1, +2, +3, +4, ...).

Calculating the Next Term

Following this pattern, to find the missing number after 218, we need to multiply 218 by 2 and then add the next number in the sequence 1, 2, 3, 4, ... which is 5.

So, the next term is calculated as:

\(\text{Next Term} = 218 \times 2 + 5\)

\(\text{Next Term} = 436 + 5\)

\(\text{Next Term} = 441\)

Verifying the Pattern and Solution

The identified pattern successfully connects each term to the next. The calculated next term is 441.

Let's summarize the steps based on the pattern:

• \(12 \times 2 + 1 = 25\)
• \(25 \times 2 + 2 = 52\)
• \(52 \times 2 + 3 = 107\)
• \(107 \times 2 + 4 = 218\)
• \(218 \times 2 + 5 = 441\)

The missing number in the series is 441.

Revision Table: Understanding Number Series

Additional Information: Strategies for Solving Number Series

Solving number series questions requires careful observation and practice. Here are some common strategies:

• Calculate Differences: Find the difference between consecutive terms. If there's no obvious pattern, find the differences of the differences, and so on.
• Look for Ratios: Check if there's a constant ratio between terms (geometric series).
• Identify Multiplication/Division Patterns: See if terms are related by multiplication or division, potentially combined with addition or subtraction.
• Check for Squares or Cubes: Terms might be squares or cubes of numbers, or related to them.
• Look for Alternating Patterns: Sometimes, the pattern alternates between two different rules.
• Combine Operations: The pattern might involve a combination of operations, as seen in this problem (multiplication and addition).

By systematically applying these strategies, you can identify the pattern and find the missing number in most series problems.