Question

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

• A. 36
• B. 40
• C. 42
• D. 56

Hint:

When the gaps between terms are not equal, look at the differences between those gaps. If that second set of differences is constant, the series follows a quadratic pattern, and it can be extended by one more step to find the missing term.

Answer 1

The Correct Option is C

To solve the problem, we need to identify the pattern in the number series and find the next number.

1. Understanding the Concepts:

- Number Series: A sequence of numbers following a certain rule.
- Pattern Recognition: Analyze the differences or formulas generating the series.

2. Given Series:

2, 6, 12, 20, 30, ?

3. Analyzing the Pattern:

Look at the pattern in the terms:

• 2 = 1 × 2
• 6 = 2 × 3
• 12 = 3 × 4
• 20 = 4 × 5
• 30 = 5 × 6

So, the nth term = \( n \times (n+1) \)

4. Calculate the Next Term:

For \( n = 6 \): \[ 6 \times 7 = 42 \]

Final Answer:

The next number in the series is 42.

Answer 2

Concept:

• When consecutive terms of a series do not have a constant difference, look at the differences of the differences (second differences). A constant second difference means the series follows a quadratic rule.
• Once the pattern in the differences is known, it can be extended by one more step to predict the next term, without needing to spot the exact algebraic formula for the nth term.

Step 1: Find the first differences between consecutive terms.
$6-2=4, \quad 12-6=6, \quad 20-12=8, \quad 30-20=10$
First differences: 4, 6, 8, 10

Step 2: Find the second differences.
$6-4=2, \quad 8-6=2, \quad 10-8=2$
The second differences are constant at 2, confirming a quadratic pattern.

Step 3: Extend the pattern by one more step.
The next first difference continues the pattern: $10+2=12$
Add this to the last term of the series: $30+12=42$

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