Series Analysis: Identifying the Pattern

First, let's simplify the terms of the given series to understand its structure:

• The first term is $ \sqrt{2} $.
• The second term is $ \sqrt{8} $, which simplifies to $ \sqrt{4 \times 2} = 2\sqrt{2} $.
• The third term is $ \sqrt{18} $, which simplifies to $ \sqrt{9 \times 2} = 3\sqrt{2} $.
• The fourth term is $ \sqrt{32} $, which simplifies to $ \sqrt{16 \times 2} = 4\sqrt{2} $.

So, the series can be rewritten as:

$ \sqrt{2} + 2\sqrt{2} + 3\sqrt{2} + 4\sqrt{2} + ..... $

We can see that this is an arithmetic progression (AP) because there is a constant difference between consecutive terms.

• The first term ($a$) of the AP is $ \sqrt{2} $.
• The common difference ($d$) is found by subtracting any term from its succeeding term, for example, $ 2\sqrt{2} - \sqrt{2} = \sqrt{2} $.

Sum Calculation for 16 Terms

The problem asks for the sum of the first 16 terms ($n=16$) of this arithmetic progression.

The formula to calculate the sum ($S_n$) of the first $n$ terms of an arithmetic progression is:

$ S_n = \frac{n}{2}[2a + (n-1)d] $

Now, we plug in the values we identified:

• $ n = 16 $
• $ a = \sqrt{2} $
• $ d = \sqrt{2} $

Substituting these values into the sum formula:

$ S_{16} = \frac{16}{2}[2(\sqrt{2}) + (16-1)(\sqrt{2})] $

Simplify the expression:

$ S_{16} = 8[2\sqrt{2} + (15)(\sqrt{2})] $

Add the terms within the bracket:

$ S_{16} = 8[2\sqrt{2} + 15\sqrt{2}] $

$ S_{16} = 8[17\sqrt{2}] $

Finally, perform the multiplication:

$ S_{16} = 136\sqrt{2} $

Thus, the sum of the first 16 terms of the series $ \sqrt{2} + \sqrt{8} + \sqrt{18} + \sqrt{32} + ..... $ is $ 136\sqrt{2} $.