Question

Find the \[100^{\text{th}}\] term of the sequence: \[\sqrt{5}, 2\sqrt{5}, 3\sqrt{5}, ..................\ .\] 

Answer 1

In the sequence:

\[3\sqrt{5} - 2\sqrt{5} = 2\sqrt{5} - \sqrt{5} = \sqrt{5}.\]

Since, the difference between consecutive terms are equal, thus the sequence is an A.P.

First term (a) \[{} = \sqrt{5}\]

Common difference (d) \[{} = \sqrt{5}\]

We know that,

⇒ \[a_n = a + (n - 1)d\]

⇒ \[a_{100} = \sqrt{5} + (100 - 1) \times \sqrt{5}\]

⇒ \[a_{100} = \sqrt{5} + 99\sqrt{5}\]

⇒ \[a_{100} = 100\sqrt{5}.\]