Understanding the Number Series Pattern

The question asks us to identify the pattern in the given number series and determine the missing number represented by the question mark (?). The series is: 9, 11, 19, 61, 239, ?.

Analyzing the Sequence Progression

Let's examine the relationship between consecutive terms in the series:

• The first term is 9.
• The second term is 11.
• The third term is 19.
• The fourth term is 61.
• The fifth term is 239.

We need to find the logic connecting these numbers to predict the sixth term.

Identifying the Underlying Mathematical Rule

Let's test different mathematical operations to find the pattern. We can observe a pattern involving multiplication and addition/subtraction:

• Term 1 to Term 2: To get from 9 to 11, we can perform: $(9 \times 1) + 2 = 11$
• Term 2 to Term 3: To get from 11 to 19, we can perform: $(11 \times 2) - 3 = 19$
• Term 3 to Term 4: To get from 19 to 61, we can perform: $(19 \times 3) + 4 = 61$
• Term 4 to Term 5: To get from 61 to 239, we can perform: $(61 \times 4) - 5 = 239$

Observing these steps, we can identify a consistent pattern:

• The multiplier increases by 1 for each subsequent step (1, 2, 3, 4, ...).
• The adjustment value also increases by 1 (2, 3, 4, 5, ...), and its sign alternates (+, -, +, -, ...).

The general formula for this pattern can be represented as:

$a_{n+1} = a_n \times n + (-1)^{n+1} \times (n+1)$

Where '$a_n$' is the current term and '$n$' represents the step number starting from 1 for the first transition (i.e., calculating the 2nd term uses n=1, calculating the 3rd term uses n=2, and so on).

Calculating the Missing Term

To find the missing term (the sixth term), we apply the identified pattern using the fifth term (239) and the next multiplier and adjustment values.

• The last multiplier used was 4 (to get from term 4 to term 5). The next multiplier is 5.
• The last adjustment was -5. The next adjustment follows the alternating sign pattern (-, +, -, +, ...), so it will be positive (+6).

Applying the rule:

Missing Term = (Previous Term) $\times$ (Next Multiplier) + (Next Adjustment)

Missing Term = $239 \times 5 + 6$

First, calculate the multiplication:

$239 \times 5 = 1195$

Now, add the adjustment:

$1195 + 6 = 1201$

Therefore, the number that should come in place of the question mark (?) is 1201.