Write the next term : `sqrt(12),sqrt(27),sqrt(48),sqrt(75)"….."`

Options

A. `sqrt(99)`
B. `sqrt(96)`
C. `sqrt(108)`
D. `sqrt(114)`

Answer

To find the next term in the sequence \( \sqrt{12}, \sqrt{27}, \sqrt{48}, \sqrt{75} \), we will first simplify each term and then identify a pattern.

### Step 1: Simplify each term
1. **First Term**:
\[ \sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3} \]
2. **Second Term**:
\[ \sqrt{27} = \sqrt{9 \times 3} = 3\sqrt{3} \]
3. **Third Term**:
\[ \sqrt{48} = \sqrt{16 \times 3} = 4\sqrt{3} \]
4. **Fourth Term**:
\[ \sqrt{75} = \sqrt{25 \times 3} = 5\sqrt{3} \]

Now, we have the simplified terms:
\[ 2\sqrt{3}, 3\sqrt{3}, 4\sqrt{3}, 5\sqrt{3} \]

### Step 2: Identify the pattern
The coefficients of \(\sqrt{3}\) in the simplified terms are:
- First term: 2
- Second term: 3
- Third term: 4
- Fourth term: 5

We can see that the coefficients form an arithmetic progression (AP) with a common difference of 1.

### Step 3: Find the next term
To find the next term, we continue the pattern:
- The next coefficient after 5 is \(5 + 1 = 6\).

Thus, the next term in the sequence will be:
\[ 6\sqrt{3} \]

### Step 4: Simplify the next term
We can express \(6\sqrt{3}\) in terms of a square root:
\[ 6\sqrt{3} = \sqrt{(6^2 \times 3)} = \sqrt{36 \times 3} = \sqrt{108} \]

### Conclusion
The next term in the sequence is:
\[ \sqrt{108} \]