Series Pattern Identification

The question asks to find the missing number in the series: 2, 10, 30, ?, 130.

Let's analyze the relationship between the term number ($n$) and the term value.

• Term 1 ($n=1$): 2
• Term 2 ($n=2$): 10
• Term 3 ($n=3$): 30
• Term 4 ($n=4$): ?
• Term 5 ($n=5$): 130

Discovering the Formula

Let's test the formula: Term $= n \times (n^2 + 1)$.

• For $n=1$: $1 \times (1^2 + 1) = 1 \times (1 + 1) = 1 \times 2 = 2$. (Matches)
• For $n=2$: $2 \times (2^2 + 1) = 2 \times (4 + 1) = 2 \times 5 = 10$. (Matches)
• For $n=3$: $3 \times (3^2 + 1) = 3 \times (9 + 1) = 3 \times 10 = 30$. (Matches)
• For $n=5$: $5 \times (5^2 + 1) = 5 \times (25 + 1) = 5 \times 26 = 130$. (Matches)

Calculating the Missing Term

The pattern holds true. Now, let's calculate the 4th term ($n=4$) using the same formula:

• For $n=4$: $4 \times (4^2 + 1) = 4 \times (16 + 1) = 4 \times 17 = 68$.

Therefore, the number that replaces the question mark is 68.