Question

Find the next term in the series:
2, 6, 12, 20, 30, ?

• A. 36
• B. 40
• C. 42
• D. 44

Hint:

Whenever you face a series where numbers grow steadily and smoothly, always scribble down the step differences above them first. Identifying a straightforward gap sequence like +4, +6, +8, +10 lets you forecast +12 immediately and solve the problem effortlessly!

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
Number series puzzles require analyzing sequential patterns to determine the logical rule governing the progression. In an increasing number sequence, the pattern is typically found by evaluating the differences between consecutive elements (arithmetic progression changes) or identifying algebraic properties such as squares or products of consecutive integers.

Step 2: Key Formula or Approach:
Identify the differences between consecutive terms: $$d_n = T_{n} - T_{n-1}$$ Determine if the difference sequence reveals a recognizable pattern, then add the next logical difference value to the final listed term to determine the missing entry.

Step 3: Detailed Explanation:
Let's find the step differences between the consecutive numbers given in the series: $6 - 2 = +4$ $12 - 6 = +6$ $20 - 12 = +8$ $30 - 20 = +10$ Notice the sequence of differences: 4, 6, 8, 10. These values represent successive even numbers, increasing by exactly $+2$ at each step. Following this logic, the next difference in the sequence must be: $$10 + 2 = +12$$ Now, add this difference value to the last known term ($30$) to locate our missing target term: $$\text{Next Term} = 30 + 12 = 42$$ Let's double-check using an alternative algebraic pattern. Each term can also be expressed as $n^2 + n$ (or $n \times (n+1)$): $1^2 + 1 = 1 \times 2 = 2$ $2^2 + 2 = 2 \times 3 = 6$ $3^2 + 3 = 3 \times 4 = 12$ $4^2 + 4 = 4 \times 5 = 20$ $5^2 + 5 = 5 \times 6 = 30$ $6^2 + 6 = 6 \times 7 = 42$ Both logical frameworks confirm that the next number in the sequence is $42$, which matches option (c).

Step 4: Final Answer:
The next term in the series is 42.