The value of `1/(sqrt(3.25) + sqrt(2.25)) + 1/(sqrt(4.25) + sqrt(3.25))+ 1/(sqrt(5.25) + sqrt(4.25)) + 1/(sqrt(6.25) + sqrt(5.25))` is:

Options

A. 1
B. 1.25
C. 1.5
D. 2.25

Answer

To solve the expression

\[ \frac{1}{\sqrt{3.25} + \sqrt{2.25}} + \frac{1}{\sqrt{4.25} + \sqrt{3.25}} + \frac{1}{\sqrt{5.25} + \sqrt{4.25}} + \frac{1}{\sqrt{6.25} + \sqrt{5.25}}, \]

we will simplify each term step by step.

### Step 1: Simplifying the first term
We start with the first term:

\[ \frac{1}{\sqrt{3.25} + \sqrt{2.25}}. \]

To simplify this, we can multiply the numerator and the denominator by the conjugate of the denominator:

\[ \frac{\sqrt{3.25} - \sqrt{2.25}}{(\sqrt{3.25} + \sqrt{2.25})(\sqrt{3.25} - \sqrt{2.25})} = \frac{\sqrt{3.25} - \sqrt{2.25}}{3.25 - 2.25} = \sqrt{3.25} - \sqrt{2.25}. \]

### Step 2: Simplifying the second term
Now, we simplify the second term:

\[ \frac{1}{\sqrt{4.25} + \sqrt{3.25}}. \]

Using the same method, we multiply by the conjugate:

\[ \frac{\sqrt{4.25} - \sqrt{3.25}}{(\sqrt{4.25} + \sqrt{3.25})(\sqrt{4.25} - \sqrt{3.25})} = \frac{\sqrt{4.25} - \sqrt{3.25}}{4.25 - 3.25} = \sqrt{4.25} - \sqrt{3.25}. \]

### Step 3: Simplifying the third term
Next, we simplify the third term:

\[ \frac{1}{\sqrt{5.25} + \sqrt{4.25}}. \]

Again, we multiply by the conjugate:

\[ \frac{\sqrt{5.25} - \sqrt{4.25}}{(\sqrt{5.25} + \sqrt{4.25})(\sqrt{5.25} - \sqrt{4.25})} = \frac{\sqrt{5.25} - \sqrt{4.25}}{5.25 - 4.25} = \sqrt{5.25} - \sqrt{4.25}. \]

### Step 4: Simplifying the fourth term
Finally, we simplify the fourth term:

\[ \frac{1}{\sqrt{6.25} + \sqrt{5.25}}. \]

Multiplying by the conjugate gives us:

\[ \frac{\sqrt{6.25} - \sqrt{5.25}}{(\sqrt{6.25} + \sqrt{5.25})(\sqrt{6.25} - \sqrt{5.25})} = \frac{\sqrt{6.25} - \sqrt{5.25}}{6.25 - 5.25} = \sqrt{6.25} - \sqrt{5.25}. \]

### Step 5: Combining all terms
Now we can combine all the simplified terms:

\[ (\sqrt{3.25} - \sqrt{2.25}) + (\sqrt{4.25} - \sqrt{3.25}) + (\sqrt{5.25} - \sqrt{4.25}) + (\sqrt{6.25} - \sqrt{5.25}). \]

Notice that the terms cancel out:

- \(\sqrt{3.25}\) cancels with \(-\sqrt{3.25}\),
- \(\sqrt{4.25}\) cancels with \(-\sqrt{4.25}\),
- \(\sqrt{5.25}\) cancels with \(-\sqrt{5.25}\).

Thus, we are left with:

\[ \sqrt{6.25} - \sqrt{2.25}. \]

### Step 6: Evaluating the final expression
Now we calculate the final values:

\[ \sqrt{6.25} = 2.5 \quad \text{and} \quad \sqrt{2.25} = 1.5. \]

So we have:

\[ 2.5 - 1.5 = 1. \]

### Final Answer
The value of the given expression is

\[ \boxed{1}. \]