18, 48, 100,180, 294, 448, ?

Options

A. 979
B. 586
C. 732
D. 648

Explanation

To solve the series: 18, 48, 100, 180, 294, 448, ?, we will analyze the pattern in the numbers step by step.

### Step 1: Identify the pattern in the series
First, let's look at the numbers in the series:
- 18
- 48
- 100
- 180
- 294
- 448

We can observe that these numbers are increasing. To find a pattern, we can express each number in terms of multiplication involving squares.

### Step 2: Express each number in the series
Let's break down each number:
- 18 can be expressed as \(2 \times 9\) (where \(9 = 3^2\))
- 48 can be expressed as \(3 \times 16\) (where \(16 = 4^2\))
- 100 can be expressed as \(4 \times 25\) (where \(25 = 5^2\))
- 180 can be expressed as \(5 \times 36\) (where \(36 = 6^2\))
- 294 can be expressed as \(6 \times 49\) (where \(49 = 7^2\))
- 448 can be expressed as \(7 \times 64\) (where \(64 = 8^2\))

### Step 3: Identify the pattern in the multipliers and squares
From the above breakdown, we can see that:
- The first factor is increasing by 1 each time: \(2, 3, 4, 5, 6, 7\)
- The second factor is the square of the next integer: \(3^2, 4^2, 5^2, 6^2, 7^2, 8^2\)

### Step 4: Predict the next number
Following the established pattern:
- The next number should be \(8 \times 9^2\)
- Calculating \(9^2\) gives us \(81\)
- Therefore, the next number is \(8 \times 81 = 648\)

### Conclusion
The missing number in the series is **648**.

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