Number Series Analysis

The problem requires finding the next number in the given numerical series: 3, 11, 27, 59.

Identifying the Series Pattern

Let's examine the relationship between consecutive terms:

• Term 1: 3
• Term 2: 11
• Term 3: 27
• Term 4: 59

Calculate the differences between terms:

• $11 - 3 = 8$
• $27 - 11 = 16$
• $59 - 27 = 32$

The differences (8, 16, 32) are doubling each time. This suggests a pattern related to multiplication by 2.

Alternatively, let's test a pattern of the form $a_{n+1} = 2 \times a_n + k$.

• For the second term: $11 = 2 \times 3 + 5 \implies 11 = 6 + 5$. So, $k=5$.
• Check for the third term: $2 \times 11 + 5 = 22 + 5 = 27$. The pattern holds.
• Check for the fourth term: $2 \times 27 + 5 = 54 + 5 = 59$. The pattern holds.

The consistent pattern is $a_{n+1} = 2 \times a_n + 5$.

Calculating the Next Term

To find the next term in the series (the fifth term), we apply the identified pattern:

Next Term = $2 \times (\text{Previous Term}) + 5$

Next Term = $2 \times 59 + 5$

Next Term = $118 + 5$

Next Term = $123$

Conclusion

The number that replaces the question mark (?) in the series is 123.