What is the simplified value of
`1/(sqrt25 - sqrt24) - 1/(sqrt24 - sqrt23) + 1/(sqrt23- sqrt22) - 1/(sqrt22 - sqrt21) + 1/(sqrt21 - sqrt20)` ?

Options

A. `5 + 2sqrt 5`
B. `5 – 5sqrt2`
C. `sqrt25 + sqrt21`
D. `sqrt""25 – 4`

Answer

To simplify the expression

\[ 1/( \sqrt{25} - \sqrt{24}) - 1/( \sqrt{24} - \sqrt{23}) + 1/( \sqrt{23} - \sqrt{22}) - 1/( \sqrt{22} - \sqrt{21}) + 1/( \sqrt{21} - \sqrt{20}), \]

we will follow these steps:

### Step 1: Rewrite Each Term
Each term in the expression has the form \( \frac{1}{\sqrt{n} - \sqrt{n-1}} \). We can rationalize each term by multiplying the numerator and denominator by \( \sqrt{n} + \sqrt{n-1} \):

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

### Step 2: Apply to Each Term
Now we apply this to each term in the expression:

1. For \( \frac{1}{\sqrt{25} - \sqrt{24}} \):
\[ = \sqrt{25} + \sqrt{24} = 5 + \sqrt{24}. \]

2. For \( -\frac{1}{\sqrt{24} - \sqrt{23}} \):
\[ = -(\sqrt{24} + \sqrt{23}). \]

3. For \( \frac{1}{\sqrt{23} - \sqrt{22}} \):
\[ = \sqrt{23} + \sqrt{22}. \]

4. For \( -\frac{1}{\sqrt{22} - \sqrt{21}} \):
\[ = -(\sqrt{22} + \sqrt{21}). \]

5. For \( \frac{1}{\sqrt{21} - \sqrt{20}} \):
\[ = \sqrt{21} + \sqrt{20}. \]

### Step 3: Combine All Terms
Now we combine all these results:

\[ (5 + \sqrt{24}) - (\sqrt{24} + \sqrt{23}) + (\sqrt{23} + \sqrt{22}) - (\sqrt{22} + \sqrt{21}) + (\sqrt{21} + \sqrt{20}). \]

### Step 4: Simplify
Now, we can simplify by canceling out terms:

- \( +\sqrt{24} \) and \( -\sqrt{24} \) cancel.
- \( -\sqrt{23} \) and \( +\sqrt{23} \) cancel.
- \( -\sqrt{22} \) and \( +\sqrt{22} \) cancel.
- \( -\sqrt{21} \) and \( +\sqrt{21} \) cancel.

After cancellation, we are left with:

\[ 5 + \sqrt{20}. \]

### Step 5: Simplify \( \sqrt{20} \)
We can further simplify \( \sqrt{20} \):

\[ \sqrt{20} = \sqrt{4 \times 5} = \sqrt{4} \cdot \sqrt{5} = 2\sqrt{5}. \]

### Final Result
Thus, the final simplified expression is:

\[ 5 + 2\sqrt{5}. \]