Analyzing the Number Series Pattern

The question asks us to identify the missing number in the sequence: 2, 12, 36, ?, 150, 252.

To solve this, we need to find a logical pattern or rule that governs the progression of these numbers.

Step-by-Step Pattern Identification

Let's denote the position of a number in the series as '$n$' and the number itself as 'Term $n$'.

• For $n=1$, Term 1 = 2
• For $n=2$, Term 2 = 12
• For $n=3$, Term 3 = 36
• For $n=4$, Term 4 = ?
• For $n=5$, Term 5 = 150
• For $n=6$, Term 6 = 252

Let's explore potential relationships. We can try expressing each term relative to its position number ($n$).

Consider the structure Term $n = n \times (\text{some factor})$.

• Term 1: $2 = 1 \times 2$
• Term 2: $12 = 2 \times 6$
• Term 3: $36 = 3 \times 12$
• Term 5: $150 = 5 \times 30$
• Term 6: $252 = 6 \times 42$

Now, let's examine the sequence formed by these factors: 2, 6, 12, ?, 30, 42.

Identifying the Pattern in the Factors

Let's find the pattern for this factor sequence based on the position '$n$'.

• For $n=1$, factor = 2
• For $n=2$, factor = 6
• For $n=3$, factor = 12
• For $n=4$, factor = ?
• For $n=5$, factor = 30
• For $n=6$, factor = 42

Let's test the pattern $n \times (n+1)$ for these factors:

• For $n=1$: $1 \times (1+1) = 1 \times 2 = 2$ (Matches)
• For $n=2$: $2 \times (2+1) = 2 \times 3 = 6$ (Matches)
• For $n=3$: $3 \times (3+1) = 3 \times 4 = 12$ (Matches)
• For $n=5$: $5 \times (5+1) = 5 \times 6 = 30$ (Matches)
• For $n=6$: $6 \times (6+1) = 6 \times 7 = 42$ (Matches)

This pattern, $n \times (n+1)$, accurately describes the factors.

Calculating the Missing Number

Using the pattern identified, the rule for the original series is:

Term $n = n \times (\text{factor for position } n)$

Term $n = n \times [n \times (n+1)]$

This formula can be simplified as: Term $n = n^2 \times (n+1) = n^3 + n^2$.

Let's use this formula to find the missing number at position $n=4$:

Term 4 = $4^3 + 4^2$

Term 4 = $64 + 16$

Term 4 = $80$

Alternatively, using the $n \times [n \times (n+1)]$ form:

Factor for $n=4$ is $4 \times (4+1) = 4 \times 5 = 20$.

Term 4 = $4 \times (\text{factor for } n=4)$

Term 4 = $4 \times 20$

Term 4 = $80$

Final Answer

The number that replaces the question mark (?) in the series 2, 12, 36, ?, 150, 252 is 80.