Determine the value of ...............
`(1)/(sqrt1 +sqrt2) +(1)/(sqrt2 + sqrt3) +(1)/(sqrt3 + sqrt4) + ……+ (1)/(sqrt120 + sqrt121)`

Options

A. `sqrt120`
B. 10
C. `12sqrt12`
D. `8`

Answer

To solve the given series

\[ S = \frac{1}{\sqrt{1} + \sqrt{2}} + \frac{1}{\sqrt{2} + \sqrt{3}} + \frac{1}{\sqrt{3} + \sqrt{4}} + \ldots + \frac{1}{\sqrt{120} + \sqrt{121}}, \]

we can simplify each term in the series using a technique that involves multiplying the numerator and denominator by the conjugate of the denominator.

### Step 1: Rewrite the terms

Each term in the series can be rewritten as follows:

\[ \frac{1}{\sqrt{n} + \sqrt{n+1}} \cdot \frac{\sqrt{n+1} - \sqrt{n}}{\sqrt{n+1} - \sqrt{n}} = \frac{\sqrt{n+1} - \sqrt{n}}{(\sqrt{n+1})^2 - (\sqrt{n})^2} = \frac{\sqrt{n+1} - \sqrt{n}}{(n+1) - n} = \sqrt{n+1} - \sqrt{n}. \]

### Step 2: Apply this to the series

Now, applying this transformation to each term in the series, we have:

\[ S = (\sqrt{2} - \sqrt{1}) + (\sqrt{3} - \sqrt{2}) + (\sqrt{4} - \sqrt{3}) + \ldots + (\sqrt{121} - \sqrt{120}). \]

### Step 3: Notice the telescoping nature

Notice that this is a telescoping series. Most terms will cancel out:

\[ S = -\sqrt{1} + \sqrt{121} = -1 + 11 = 10. \]

### Conclusion

Thus, the value of the series is

\[ \boxed{10}. \]

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