Pattern Analysis

The pattern in this series can be found by looking at the differences between consecutive terms, or by observing which numbers are skipped.

Method 1: Analyzing the Differences

1. 
   Find the Differences:
   

   The sequence of increments between the numbers is:

   
   1, 1, 2, 1, 1, 2, 1, 1, ...
   


2. 
   Identify the Repeating Pattern:
   

   The differences follow a repeating cycle of (+1, +1, +2).

   
   
   
   
   • 1 (+1) → 2 (+1) → 3
   
   
   • 3 (+2) → 5
   
   
   • 5 (+1) → 6 (+1) → 7
   
   
   • and so on...
   
   
   
   


3. 
   Apply the Pattern to Find the Next Term:
   

   The series just completed two "+1" steps (from 9 to 11). The next step in the cycle is "+2".

   
   11 + 2 = 13
   

Method 2: Identifying Skipped Numbers

A simpler way to view the series is as a list of all positive integers with the multiples of 4 removed:

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, ...

After 11, the next number to be skipped is 12. Therefore, the next term in the series is 13.

Final Answer: The correct answer is (D) 13.