Question

Find next number: 2, 6, 12, 20, 30, ?

• A. 36
• B. 40
• C. 42
• D. 44

Hint:

If a sequence grows slowly, always check the "difference of differences" first. If the first set of differences is $4, 6, 8, 10$, the second set of differences is a constant $+2$, indicating a quadratic relationship ($n^2$).

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
This is a number series problem where each term follows a specific mathematical logic or pattern. To find the missing number, we need to identify the relationship between consecutive terms, often by looking at their differences or using algebraic formulas.

Step 2: Key Formula or Approach:
We can analyze the differences between consecutive terms:
• $d_1 = 6 - 2$
• $d_2 = 12 - 6$
• $d_3 = 20 - 12$
• $d_4 = 30 - 20$ Alternatively, we can look for a square-based pattern like $n^2 + n$.

Step 3: Detailed Explanation:
Using the difference method:
• $6 - 2 = 4$
• $12 - 6 = 6$
• $20 - 12 = 8$
• $30 - 20 = 10$ The differences are $4, 6, 8, 10$, which are consecutive even numbers. The next difference must be $12$. \[ \text{Next number} = 30 + 12 = 42 \] Using the $n^2 + n$ method:
• $1^2 + 1 = 2$
• $2^2 + 2 = 6$
• $3^2 + 3 = 12$
• $4^2 + 4 = 20$
• $5^2 + 5 = 30$
• $6^2 + 6 = 36 + 6 = 42$ Both methods confirm the same result.

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