? 16 28 58 114 204

Options

A. 7
B. 9
C. 14
D. 6

Explanation

To solve the series: 16, 28, 58, 114, 204, we need to identify the pattern in the differences between the consecutive terms. Let's break it down step by step.

### Step 1: Find the differences between consecutive terms
- The difference between the first two terms:
\( 28 - 16 = 12 \)

- The difference between the second and third terms:
\( 58 - 28 = 30 \)

- The difference between the third and fourth terms:
\( 114 - 58 = 56 \)

- The difference between the fourth and fifth terms:
\( 204 - 114 = 90 \)

### Step 2: List the differences
Now we have the differences:
- 12, 30, 56, 90

### Step 3: Find the differences of the differences
Next, we will find the differences of these differences:
- The difference between the first two differences:
\( 30 - 12 = 18 \)

- The difference between the second and third differences:
\( 56 - 30 = 26 \)

- The difference between the third and fourth differences:
\( 90 - 56 = 34 \)

### Step 4: List the second differences
Now we have the second differences:
- 18, 26, 34

### Step 5: Find the differences of the second differences
Now, we will find the differences of these second differences:
- The difference between the first two second differences:
\( 26 - 18 = 8 \)

- The difference between the second and third second differences:
\( 34 - 26 = 8 \)

### Step 6: Identify the pattern
The third differences are constant (8), indicating that the original series is generated by a polynomial of degree 3.

### Step 7: Predict the next difference
Since the last second difference was 34, the next second difference will be:
\( 34 + 8 = 42 \)

### Step 8: Calculate the next first difference
Now we can find the next first difference:
\( 90 + 42 = 132 \)

### Step 9: Calculate the next term in the series
Finally, we can find the next term in the series:
\( 204 + 132 = 336 \)

Thus, the next number in the series is **336**.

### Summary of Steps
1. Find the differences between consecutive terms.
2. List the differences.
3. Find the differences of the differences (second differences).
4. List the second differences.
5. Find the differences of the second differences (third differences).
6. Identify the pattern in the differences.
7. Predict the next difference.
8. Calculate the next first difference.
9. Calculate the next term in the series.