What will come in place of the question mark (?) in the following number series?
4 29 129 354 754 (?)

Options

A. 1376
B. 1368
C. 1379
D. 1739

Answer

To find the missing number in the series: 4, 29, 129, 354, 754, (?), we will analyze the differences between the consecutive terms.

### Step 1: Calculate the differences between consecutive terms
- The difference between 29 and 4:
\[ 29 - 4 = 25 \]
- The difference between 129 and 29:
\[ 129 - 29 = 100 \]
- The difference between 354 and 129:
\[ 354 - 129 = 225 \]
- The difference between 754 and 354:
\[ 754 - 354 = 400 \]

### Step 2: List the differences
The differences we calculated are:
- 25
- 100
- 225
- 400

### Step 3: Identify the pattern in the differences
Now, let's observe the differences:
- The first difference is \(25\)
- The second difference is \(100\) (which is \(25 \times 4\))
- The third difference is \(225\) (which is \(15^2\) or \(25 + 100 + 100\))
- The fourth difference is \(400\) (which is \(20^2\) or \(25 + 100 + 225\))

### Step 4: Calculate the next difference
The differences appear to increase in a specific pattern. Let's calculate the next difference:
- The difference between the differences:
- \(100 - 25 = 75\)
- \(225 - 100 = 125\)
- \(400 - 225 = 175\)

The differences of the differences are:
- 75
- 125
- 175

### Step 5: Find the next difference of the differences
The next difference can be calculated by adding 50 to the last difference:
\[ 175 + 50 = 225 \]

### Step 6: Calculate the next difference
Now, we add this new difference to the last difference:
\[ 400 + 225 = 625 \]

### Step 7: Calculate the missing number
Finally, we add this difference to the last number in the series:
\[ 754 + 625 = 1379 \]

Thus, the missing number in the series is **1379**.

### Summary of the Solution:
The missing number in the series is **1379**.