Step 1: Understanding the Concept:
A number sequence or series is a collection of numbers that follows a specific logical pattern.
To solve sequence problems, we must identify the relationship between consecutive terms.
Common patterns include constant addition (Arithmetic Progression), constant multiplication (Geometric Progression), squares, cubes, or alternating operations.
Key Formula or Approach:
We examine the ratio between consecutive terms (\( \frac{a_{n+1}}{a_n} \)):
First term to Second: \( 6 \div 3 = 2 \)
Second term to Third: \( 12 \div 6 = 2 \)
Third term to Fourth: \( 24 \div 12 = 2 \)
This indicates a Geometric Progression where each term is obtained by multiplying the previous term by a common ratio \( r = 2 \).
Step 2: Detailed Explanation:
Let's list the operations performed to obtain each term:
Term 1 = 3
Term 2 = \( 3 \times 2 = 6 \)
Term 3 = \( 6 \times 2 = 12 \)
Term 4 = \( 12 \times 2 = 24 \)
We can see a clear pattern of "doubling" the previous number.
To find the next (fifth) term, we simply apply the same logic of multiplying by 2 to the fourth term.
Next Term = \( \text{Fourth Term} \times 2 \)
Next Term = \( 24 \times 2 \)
\[ 24 \times 2 = 48 \]
Alternatively, we can look at the differences:
6 - 3 = 3
12 - 6 = 6
24 - 12 = 12
The difference is also doubling (3, 6, 12). The next difference should be \( 12 \times 2 = 24 \).
Next Term = \( 24 + 24 = 48 \).
Both logical paths lead to the same result.
Step 3: Final Answer:
The next term in the sequence is 48.
This matches Option (C).