Question

Find the missing number in the series: \(2, 6, 12, 20, 30, ?\)

• A. \(38\)
• B. \(40\)
• C. \(42\)
• D. \(44\)

Hint:

When numbers increase quickly in a series, checking the differences between consecutive terms often reveals the pattern.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
We are given a sequence of numbers and must identify the underlying mathematical pattern to predict the next number in the series.
Step 2: Detailed Solution:
One of the most effective ways to analyze a number series is to look at the differences between consecutive terms.
Let's calculate the primary differences:
\(6 - 2 = 4\)
\(12 - 6 = 6\)
\(20 - 12 = 8\)
\(30 - 20 = 10\)
The sequence of differences is \(4, 6, 8, 10\).
We can clearly observe that these differences form an Arithmetic Progression themselves, increasing by \(2\) each time.
Following this established pattern, the next difference must be:
\(10 + 2 = 12\)
To find the missing number, we add this next difference to the last known term of the series:
\(30 + 12 = 42\)
(Alternative Logic: The sequence can also be seen as \(n^2 + n\) where \(n=1,2,3,...\). For \(n=6\), \(6^2 + 6 = 36 + 6 = 42\). Both logics yield the same result.)
Step 3: Final Answer:
The missing number is \(42\).