Let `f (n)=(4n + sqrt(4n ^(2) -1))/( sqrt(2n +1 )+sqrt(2n-1)),n in N` then the remainder when `f (1) + f (2) + f (3) + ..... + f (60)` is divided by 9 is.

Answer

To solve the problem, we need to evaluate the function \( f(n) = \frac{4n + \sqrt{4n^2 - 1}}{\sqrt{2n + 1} + \sqrt{2n - 1}} \) for \( n = 1, 2, \ldots, 60 \) and find the remainder when the sum \( f(1) + f(2) + \ldots + f(60) \) is divided by 9.

### Step 1: Simplifying \( f(n) \)

We start with the function:

\[ f(n) = \frac{4n + \sqrt{4n^2 - 1}}{\sqrt{2n + 1} + \sqrt{2n - 1}} \]

We can rewrite \( 4n \) as \( 2n + 2n \) and \( \sqrt{4n^2 - 1} \) as \( \sqrt{(2n)^2 - 1^2} \).

### Step 2: Rationalizing the Denominator

We can rationalize the denominator:

\[ f(n) = \frac{(4n + \sqrt{4n^2 - 1})(\sqrt{2n + 1} - \sqrt{2n - 1})}{(\sqrt{2n + 1} + \sqrt{2n - 1})(\sqrt{2n + 1} - \sqrt{2n - 1})} \]

The denominator simplifies to:

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

### Step 3: Evaluating the Numerator

Now, we simplify the numerator:

\[ (4n + \sqrt{4n^2 - 1})(\sqrt{2n + 1} - \sqrt{2n - 1}) = (4n)(\sqrt{2n + 1} - \sqrt{2n - 1}) + \sqrt{4n^2 - 1}(\sqrt{2n + 1} - \sqrt{2n - 1}) \]

### Step 4: Finding a Pattern

After simplifying, we can see that:

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

This leads us to:

\[ f(n) = \frac{1}{2} \left( (2n + 1)^{3/2} - (2n - 1)^{3/2} \right) \]

### Step 5: Summing \( f(n) \)

Now we need to compute:

\[ S = f(1) + f(2) + \ldots + f(60) \]

Using the telescoping nature of the series, we find that:

\[ S = \frac{1}{2} \left( (121^{3/2} - 1^{3/2}) + (119^{3/2} - 3^{3/2}) + \ldots + (1^{3/2} - 119^{3/2}) \right) \]

### Step 6: Final Calculation

The terms will cancel out, and we are left with:

\[ S = \frac{1}{2} \left( 121^{3/2} - 1^{3/2} \right) = \frac{1}{2} (1331 - 1) = \frac{1330}{2} = 665 \]

### Step 7: Finding the Remainder

Finally, we need to find the remainder when \( 665 \) is divided by \( 9 \):

\[ 665 \div 9 = 73 \quad \text{remainder } 8 \]

Thus, the final answer is:

\[ \text{The remainder when } f(1) + f(2) + \ldots + f(60) \text{ is divided by } 9 \text{ is } 8. \]