6, 7, 16, 51, 208, ?

Options

A. 1036
B. 1045
C. 1048
D. 1056

Explanation

To find the missing number in the series: 6, 7, 16, 51, 208, ?, we can analyze the pattern of the series step by step.

### Step-by-Step Solution:

1. **Identify the Pattern:**
- The series starts with 6, then 7, followed by 16, 51, and 208.
- We notice that the numbers are increasing, suggesting a possible multiplication and addition pattern.

2. **Calculate the First Transition (6 to 7):**
- From 6 to 7, we can see that:
\[ 6 \times 1 + 1 = 7 \]
- Here, we multiplied by 1 and added 1.

3. **Calculate the Second Transition (7 to 16):**
- From 7 to 16, we can see that:
\[ 7 \times 2 + 2 = 16 \]
- Here, we multiplied by 2 and added 2.

4. **Calculate the Third Transition (16 to 51):**
- From 16 to 51, we can see that:
\[ 16 \times 3 + 3 = 51 \]
- Here, we multiplied by 3 and added 3.

5. **Calculate the Fourth Transition (51 to 208):**
- From 51 to 208, we can see that:
\[ 51 \times 4 + 4 = 208 \]
- Here, we multiplied by 4 and added 4.

6. **Calculate the Fifth Transition (208 to ?):**
- Following the established pattern, we will multiply by 5 and add 5:
\[ 208 \times 5 + 5 \]
- First, calculate \(208 \times 5\):
\[ 208 \times 5 = 1040 \]
- Now, add 5:
\[ 1040 + 5 = 1045 \]

### Conclusion:
The missing number in the series is **1045**.