72, 193, 293, 374, 438, 487, ?

Options

A. 521
B. 523
C. 525
D. 527

Explanation

To solve the series: 72, 193, 293, 374, 438, 487, ?, we will follow these steps:

### Step 1: Identify the series
The series given is: 72, 193, 293, 374, 438, 487.

### Step 2: Calculate the differences between consecutive terms
Let's find the difference between each pair of consecutive terms:

1. **Difference between 193 and 72:**
\[ 193 - 72 = 121 \]

2. **Difference between 293 and 193:**
\[ 293 - 193 = 100 \]

3. **Difference between 374 and 293:**
\[ 374 - 293 = 81 \]

4. **Difference between 438 and 374:**
\[ 438 - 374 = 64 \]

5. **Difference between 487 and 438:**
\[ 487 - 438 = 49 \]

Now we have the differences: 121, 100, 81, 64, 49.

### Step 3: Analyze the differences
The differences we calculated are:
- 121 (which is \(11^2\))
- 100 (which is \(10^2\))
- 81 (which is \(9^2\))
- 64 (which is \(8^2\))
- 49 (which is \(7^2\))

### Step 4: Identify the pattern
The differences are perfect squares decreasing by 1:
- \(11^2, 10^2, 9^2, 8^2, 7^2\)

Following this pattern, the next difference should be:
- \(6^2 = 36\)

### Step 5: Calculate the next term
To find the missing number, we add the next difference (36) to the last term in the series (487):
\[ 487 + 36 = 523 \]

### Conclusion
The missing number in the series is **523**.