The value of `(1-sqrt2) + (sqrt2-sqrt3)+(sqrt3-sqrt4)+ ............ + (sqrt15-sqrt16)` 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 follow these steps:

### Step 1: Write the expression clearly
The expression can be rewritten as:
\[ 1 - \sqrt{2} + \sqrt{2} - \sqrt{3} + \sqrt{3} - \sqrt{4} + \ldots + \sqrt{15} - \sqrt{16} \]

### Step 2: Group the terms
Notice that the terms can be grouped in a way that many terms will cancel out:
\[ (1 - \sqrt{2}) + (\sqrt{2} - \sqrt{3}) + (\sqrt{3} - \sqrt{4}) + \ldots + (\sqrt{15} - \sqrt{16}) \]

### Step 3: Cancel out terms
When we group and rearrange the terms, we see that:
- The \( -\sqrt{2} \) from the first term cancels with the \( +\sqrt{2} \) from the second term.
- The \( -\sqrt{3} \) from the second term cancels with the \( +\sqrt{3} \) from the third term.
- This pattern continues until the last term.

Thus, we are left with:
\[ 1 - \sqrt{16} \]

### Step 4: Calculate \(\sqrt{16}\)
Now, we calculate \(\sqrt{16}\):
\[ \sqrt{16} = 4 \]

### Step 5: Substitute back into the expression
Substituting back, we get:
\[ 1 - 4 = -3 \]

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

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