What is the value of `(5)/(sqrt(2)+1)+(5)/(sqrt(3)+sqrt(2))+(5)/(sqrt(4)+sqrt(3))+….(5)/(sqrt(121)+sqrt(120))` ?

Options

A. 25
B. 50
C. 100
D. 75

Answer

To solve the expression \( \frac{5}{\sqrt{2}+1} + \frac{5}{\sqrt{3}+\sqrt{2}} + \frac{5}{\sqrt{4}+\sqrt{3}} + \ldots + \frac{5}{\sqrt{121}+\sqrt{120}} \), we will simplify each term step by step.

### Step 1: Simplify the first term
The first term is \( \frac{5}{\sqrt{2}+1} \). To simplify it, we will rationalize the denominator:

\[ \frac{5}{\sqrt{2}+1} \cdot \frac{\sqrt{2}-1}{\sqrt{2}-1} = \frac{5(\sqrt{2}-1)}{(\sqrt{2})^2 - (1)^2} = \frac{5(\sqrt{2}-1)}{2-1} = 5(\sqrt{2}-1) = 5\sqrt{2} - 5 \]

**Hint:** Rationalizing the denominator helps eliminate the square root from the denominator.

### Step 2: Simplify the second term
The second term is \( \frac{5}{\sqrt{3}+\sqrt{2}} \). We will rationalize it as well:

\[ \frac{5}{\sqrt{3}+\sqrt{2}} \cdot \frac{\sqrt{3}-\sqrt{2}}{\sqrt{3}-\sqrt{2}} = \frac{5(\sqrt{3}-\sqrt{2})}{(\sqrt{3})^2 - (\sqrt{2})^2} = \frac{5(\sqrt{3}-\sqrt{2})}{3-2} = 5(\sqrt{3}-\sqrt{2}) = 5\sqrt{3} - 5\sqrt{2} \]

**Hint:** Use the difference of squares to simplify the expression when rationalizing.

### Step 3: Simplify the third term
The third term is \( \frac{5}{\sqrt{4}+\sqrt{3}} \):

\[ \frac{5}{\sqrt{4}+\sqrt{3}} \cdot \frac{\sqrt{4}-\sqrt{3}}{\sqrt{4}-\sqrt{3}} = \frac{5(\sqrt{4}-\sqrt{3})}{(\sqrt{4})^2 - (\sqrt{3})^2} = \frac{5(\sqrt{4}-\sqrt{3})}{4-3} = 5(\sqrt{4}-\sqrt{3}) = 5(2-\sqrt{3}) = 10 - 5\sqrt{3} \]

**Hint:** Each term can be simplified similarly by rationalizing the denominator.

### Step 4: Generalize the pattern
Continuing this process, we can see that each term \( \frac{5}{\sqrt{n} + \sqrt{n-1}} \) simplifies to:

\[ \frac{5(\sqrt{n}-\sqrt{n-1})}{n - (n-1)} = 5(\sqrt{n}-\sqrt{n-1}) \]

### Step 5: Write the entire sum
The entire sum can be expressed as:

\[ 5(\sqrt{2}-1) + 5(\sqrt{3}-\sqrt{2}) + 5(\sqrt{4}-\sqrt{3}) + \ldots + 5(\sqrt{121}-\sqrt{120}) \]

### Step 6: Notice the telescoping nature
Notice that this is a telescoping series. Most terms will cancel out:

\[ = 5(\sqrt{121} - 1) = 5(11 - 1) = 5 \times 10 = 50 \]

### Final Answer
Thus, the value of the entire expression is:

\[ \boxed{50} \]