`(1-1/(2^2))(1-1/(3^2))......(1-1/(9^2))(1-1/(10^2))=?`

Options

A. '5/12'
B. `1/2`
C. `11/20`
D. `7/10`

Answer

To solve the expression \((1 - \frac{1}{2^2})(1 - \frac{1}{3^2}) \cdots (1 - \frac{1}{9^2})(1 - \frac{1}{10^2})\), we can break it down step by step.

### Step 1: Rewrite Each Term
Each term in the product can be rewritten as follows:
\[ 1 - \frac{1}{n^2} = \frac{n^2 - 1}{n^2} \]
This can be factored using the difference of squares:
\[ n^2 - 1 = (n - 1)(n + 1) \]
Thus, we can express each term as:
\[ 1 - \frac{1}{n^2} = \frac{(n - 1)(n + 1)}{n^2} \]

### Step 2: Apply to All Terms
Now we apply this to each term in the product from \(n = 2\) to \(n = 10\):
\[ (1 - \frac{1}{2^2})(1 - \frac{1}{3^2}) \cdots (1 - \frac{1}{10^2}) = \frac{(1)(3)}{2^2} \cdot \frac{(2)(4)}{3^2} \cdots \frac{(9)(11)}{10^2} \]

### Step 3: Write the Full Expression
The full expression becomes:
\[ \frac{(1 \cdot 3)(2 \cdot 4)(3 \cdot 5)(4 \cdot 6)(5 \cdot 7)(6 \cdot 8)(7 \cdot 9)(8 \cdot 10)(9 \cdot 11)}{2^2 \cdot 3^2 \cdot 4^2 \cdot 5^2 \cdot 6^2 \cdot 7^2 \cdot 8^2 \cdot 9^2 \cdot 10^2} \]

### Step 4: Simplify the Numerator and Denominator
The numerator can be simplified as:
\[ (1 \cdot 2 \cdot 3 \cdots 9) \cdot (3 \cdot 4 \cdots 11) \]
The denominator is:
\[ (2 \cdot 3 \cdots 10)^2 \]

### Step 5: Cancel Out Terms
Notice that many terms will cancel out:
- The \(2\) from \(2^2\) cancels with \(2\) in the numerator.
- The \(3\) from \(3^2\) cancels with \(3\) in the numerator, and so on.

After canceling, we are left with:
\[ \frac{1 \cdot 11}{10 \cdot 10} = \frac{11}{100} \]

### Step 6: Final Calculation
The final answer is:
\[ \frac{11}{20} \]

### Conclusion
Thus, the value of the expression \((1 - \frac{1}{2^2})(1 - \frac{1}{3^2}) \cdots (1 - \frac{1}{10^2})\) is \(\frac{11}{20}\).