Step 1: Understanding the Concept:
Number series puzzles require us to identify a hidden logical rule or pattern that governs the progression of numbers. In competitive exams, these patterns are usually arithmetic (differences), geometric (ratios), or based on squares and cubes.
When a series increases steadily but not explosively, the pattern is most likely found in the "differences" between consecutive terms. This is known as analyzing the "first-order" difference. If those differences themselves follow a pattern, it is a "second-order" arithmetic progression.
Step 2: Key Formula or Approach:
1. Identify the differences between each pair of consecutive terms: \(d_n = T_n - T_{n-1}\).
2. Look for a recognizable sequence in those differences (e.g., even numbers, prime numbers, or a constant increase).
3. Add the next logical difference to the last term provided in the series.
Step 3: Detailed Explanation:
Let's calculate the step differences for the given numbers: 2, 6, 12, 20, 30.
Difference 1: \(6 - 2 = +4\)
Difference 2: \(12 - 6 = +6\)
Difference 3: \(20 - 12 = +8\)
Difference 4: \(30 - 20 = +10\)
Now, let's examine the resulting sequence of differences: 4, 6, 8, 10.
It is obvious that these values are consecutive even numbers, increasing by exactly \(+2\) at each step. Following this logic, the next difference in the sequence must be:
\[ 10 + 2 = +12 \]
To find the missing term, we add this next logical difference (12) to the last known term in the series (30):
\[ \text{Next Term} = 30 + 12 = 42 \]
As a double-check, we can look for an alternative algebraic pattern. Notice that each term can be expressed as \(n^2 + n\) or \(n \times (n+1)\):
\(1^2 + 1 = 1 \times 2 = 2\)
\(2^2 + 2 = 2 \times 3 = 6\)
\(3^2 + 3 = 3 \times 4 = 12\)
\(4^2 + 4 = 4 \times 5 = 20\)
\(5^2 + 5 = 5 \times 6 = 30\)
The next term would be \(6^2 + 6 = 6 \times 7 = 42\). Both logical frameworks confirm the result.
Step 4: Final Answer:
The next term in the series is 42, which corresponds to option (c).