Sequence And Series

Step 1: Identify the sequence by computing the terms step-by-step using the given recurrence relation with initial terms and .
Step 2: Calculate the terms:
- ,
- ,
- ,
- .
Step 3: Identify the 6th term. The 6th term is .
Step 4: Verify the sequence.
The sequence is 1, 2, 3, 5, 8, 13, which matches the Fibonacci-like pattern, confirming the calculation.

The given series is 2, 6, 12, 20. Let's look at the differences between successive terms: The differences are increasing by 2 each time, so the next difference will be 10. Thus, the next term in the series will be: Thus, the correct answer is 30.