Question

Sum of n terms of the series \[\sqrt{2} + \sqrt{8} + \sqrt{18} + \sqrt{32} +\] .......  is

Options

• \[\frac{n (n + 1)}{2}\]

• 2n (n + 1)

• \[\frac{n (n + 1)}{\sqrt{2}}\]

• 1

Answer 1

\[\frac{n (n + 1)}{\sqrt{2}}\] 

Let \[T_n\] be the nth term of the given series.
Thus, we have

\[T_n = \sqrt{2 \times n^2} = n\sqrt{2}\]

Now, let

\[S_n\] be the sum of n terms of the given series.
Thus, we have:

\[S_n = \sqrt{2} \sum^n_{k = 1} \left( k \right)\]

\[ \Rightarrow S_n = \sqrt{2}\left[ \frac{n\left( n + 1 \right)}{2} \right]\]

\[ \Rightarrow S_n = \frac{n\left( n + 1 \right)}{\sqrt{2}}\]