This question asks us to find the missing number in the given series: 3, 7, 13, 27, 53, ?

To solve a number series question, we need to identify the pattern that connects consecutive terms.

Analyzing the Number Series Pattern

Let's look at the difference between consecutive terms:

• 7 - 3 = 4
• 13 - 7 = 6
• 27 - 13 = 14
• 53 - 27 = 26

The differences (4, 6, 14, 26) do not follow a simple arithmetic or geometric progression directly. This suggests that the pattern might involve more than just addition or subtraction.

Let's try to find a relationship between each term and the previous term using multiplication and addition/subtraction.

• From 3 to 7: We can see that \(3 \times 2 = 6\), and \(6 + 1 = 7\). So, the pattern could be \( \times 2 + 1 \).
• From 7 to 13: If we apply \( \times 2 + 1 \), we get \(7 \times 2 + 1 = 14 + 1 = 15\), which is not 13. Let's try \( \times 2 - 1 \). We get \(7 \times 2 - 1 = 14 - 1 = 13\). This matches.

It appears the pattern might be alternating between \( \times 2 + 1 \) and \( \times 2 - 1 \).

Applying the Alternating Pattern

Let's verify this alternating pattern for the entire series:

The pattern successfully generates all the given terms in the series. The pattern alternates between multiplying the previous term by 2 and adding 1, and multiplying the previous term by 2 and subtracting 1.

The sequence of operations is: Start, \( \times 2 + 1 \), \( \times 2 - 1 \), \( \times 2 + 1 \), \( \times 2 - 1 \), ...

The last operation applied to get 53 was \( \times 2 - 1 \). Therefore, to find the missing number, we must apply the next operation in the sequence, which is \( \times 2 + 1 \), to the last term, 53.

Calculating the Missing Number

The missing number is the term after 53. We apply the pattern \( \times 2 + 1 \) to 53:

\( \text{Missing Number} = 53 \times 2 + 1 \)

\( \text{Missing Number} = 106 + 1 \)

\( \text{Missing Number} = 107 \)

Thus, the missing number in the series 3, 7, 13, 27, 53, ? is 107.

Revision Table: Number Series Patterns

Additional Information: Solving Number Series Problems

Here are some tips for solving number series problems:

• Look for simple patterns first: constant difference, constant ratio.
• Examine the differences between consecutive terms. If they form a pattern, identify it.
• Look for patterns involving multiplication and addition/subtraction relating terms.
• Check for alternating patterns (e.g., different operations applied to alternate terms or alternating operations applied sequentially).
• Consider square numbers, cube numbers, prime numbers, or Fibonacci sequences if other patterns are not obvious.
• Practice different types of series to improve pattern recognition skills.