The value of `(1-sqrt(2))+(sqrt(2)+sqrt(3))-(sqrt(3)-sqrt(4))-...+(sqrt(15)-sqrt(16))` is

Options

A. `0`
B. `1`
C. `-3`
D. `4`

Answer

To solve the expression \((1 - \sqrt{2}) + (\sqrt{2} + \sqrt{3}) - (\sqrt{3} - \sqrt{4}) - \ldots + (\sqrt{15} - \sqrt{16})\), we can simplify it step by step.

### Step 1: Write out the series clearly
The expression can be rewritten to show the pattern more clearly:
\[ 1 - \sqrt{2} + \sqrt{2} + \sqrt{3} - \sqrt{3} - \sqrt{4} + \sqrt{4} + \sqrt{5} - \sqrt{5} - \sqrt{6} + \ldots + \sqrt{15} - \sqrt{16} \]

### Step 2: Identify and cancel out terms
Notice that many terms will cancel each other out:
- \(-\sqrt{2} + \sqrt{2} = 0\)
- \(-\sqrt{3} + \sqrt{3} = 0\)
- \(-\sqrt{4} + \sqrt{4} = 0\)
- \(-\sqrt{5} + \sqrt{5} = 0\)
- \(-\sqrt{6} + \sqrt{6} = 0\)
- This pattern continues up to \(-\sqrt{15} + \sqrt{15} = 0\)

### Step 3: Identify the remaining terms
After cancellation, the only terms that do not get canceled are:
\[ 1 - \sqrt{16} \]

### Step 4: Simplify the remaining expression
Now we can simplify:
\[ 1 - \sqrt{16} = 1 - 4 = -3 \]

### Conclusion
Thus, the value of the entire expression is:
\[ \boxed{-3} \]