Question

For the given numbers \(\sqrt{3}, \sqrt{12}, \sqrt{27}, \sqrt{48}, \cdots\)

Assertion (A): To find whether these terms form an A.P. or not. Express each term as the product of a natural number and \(\sqrt{3}\)
i.e. \(\sqrt{3}=1 \times \sqrt{3}, \sqrt{12}=2 \sqrt{3}\),
\(\sqrt{27}=3 \sqrt{3}, \sqrt{48}=4 \sqrt{3}\), etc.

Reason (R): Since, for the given numbers difference between the consecutive terms is same. It is an A.P.

Options

• A is true, R is false.

• A is false, R is true.

• Both A and R are true and R is the correct reason for A .

• Both A and R are true and R is the incorrect reason for A.

Answer 1

Both A and R are true and R is the correct reason for A .

Explanation:

Given, sequence

⇒ \[\sqrt{3}, \sqrt{12}, \sqrt{27}, \sqrt{48}............\]

⇒ \[\sqrt{3}, \sqrt{4 \times 3}, \sqrt{9 \times 3}, \sqrt{16 \times 3}............\]

⇒ \[\sqrt{3}, \sqrt{4} \times \sqrt{3}, \sqrt{9} \times \sqrt{3}, \sqrt{16} \times \sqrt{3}............\]

⇒ \[\sqrt{3}, 2\sqrt{3}, 3\sqrt{3}, 4\sqrt{3}............\]

Difference between first term and second term \[{} = 2\sqrt{3} - \sqrt{3} = \sqrt{3}\]

Difference between second term and third term \[{} = 3\sqrt{3} - 2\sqrt{3} = \sqrt{3}\]

So, the first term \[{} = \sqrt{3}\], common difference \[{} = \sqrt{3}\]

So, assertion is true.

Since, for the given number difference between the consecutive term is same. Its an A.P., means reason is true and it clearly explain assertion.