The sum to n terms of the series `(1)/(sqrt(7)+sqrt(10))+(1)/(sqrt(10)+sqrt(13))+(1)/(sqrt(13)+sqrt(16))+...`is

Options

A. `(1)/(3) (sqrt(7+3n)-sqrt(7))`
B. `(sqrt(4+3n)-2)/(3)`
C. `(1)/(3)(sqrt(10+3n)-sqrt(10))`
D. none of these

Answer

To find the sum of the series given by

\[ S_n = \frac{1}{\sqrt{7} + \sqrt{10}} + \frac{1}{\sqrt{10} + \sqrt{13}} + \frac{1}{\sqrt{13} + \sqrt{16}} + \ldots \]

we start by identifying the general term of the series.

### Step 1: Identify the General Term

The general term \( T_r \) of the series can be expressed as:

\[ T_r = \frac{1}{\sqrt{3r + 4} + \sqrt{3r + 7}} \]

where \( r \) starts from 1 and goes up to \( n \).

### Step 2: Rationalize the Denominator

To simplify \( T_r \), we will rationalize the denominator. We multiply the numerator and denominator by the conjugate of the denominator:

\[ T_r = \frac{\sqrt{3r + 7} - \sqrt{3r + 4}}{(\sqrt{3r + 4} + \sqrt{3r + 7})(\sqrt{3r + 7} - \sqrt{3r + 4})} \]

The denominator simplifies to:

\[ (\sqrt{3r + 7})^2 - (\sqrt{3r + 4})^2 = (3r + 7) - (3r + 4) = 3 \]

Thus, we have:

\[ T_r = \frac{\sqrt{3r + 7} - \sqrt{3r + 4}}{3} \]

### Step 3: Write the Sum of the Series

Now we can write the sum \( S_n \):

\[ S_n = \sum_{r=1}^{n} T_r = \sum_{r=1}^{n} \frac{\sqrt{3r + 7} - \sqrt{3r + 4}}{3} \]

### Step 4: Factor Out the Constant

We can factor out the constant \( \frac{1}{3} \):

\[ S_n = \frac{1}{3} \sum_{r=1}^{n} (\sqrt{3r + 7} - \sqrt{3r + 4}) \]

### Step 5: Telescoping Series

Notice that this is a telescoping series. When we expand the sum, we will see that most terms cancel out:

\[ S_n = \frac{1}{3} \left( (\sqrt{3 \cdot 1 + 7} - \sqrt{3 \cdot 1 + 4}) + (\sqrt{3 \cdot 2 + 7} - \sqrt{3 \cdot 2 + 4}) + \ldots + (\sqrt{3n + 7} - \sqrt{3n + 4}) \right) \]

The first term of the first square root and the last term of the last square root will remain. Thus, we have:

\[ S_n = \frac{1}{3} \left( \sqrt{3n + 7} - \sqrt{4} \right) \]

### Step 6: Final Expression

Since \( \sqrt{4} = 2 \), we can write:

\[ S_n = \frac{1}{3} (\sqrt{3n + 7} - 2) \]

### Summary

The sum to \( n \) terms of the series is:

\[ S_n = \frac{1}{3} (\sqrt{3n + 7} - 2) \]