Find the value of
`(1-1/(2^(2)))(1-1/(3^(2)))(1-1/(4^(2)))(1-1/(5^(2)))…….(1-1/(9^(2)))(1-1/(10^(2)))`

Options

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

Answer

To find the value of the expression

\[ (1 - \frac{1}{2^2})(1 - \frac{1}{3^2})(1 - \frac{1}{4^2})(1 - \frac{1}{5^2}) \ldots (1 - \frac{1}{9^2})(1 - \frac{1}{10^2}), \]

we will simplify each term and then find the product.

### Step 1: Simplify Each Term
Each term in the expression can be rewritten as follows:

\[ 1 - \frac{1}{n^2} = \frac{n^2 - 1}{n^2} = \frac{(n - 1)(n + 1)}{n^2}. \]

### Step 2: Write the Entire Expression
Now, we can write the entire expression using the simplified form:

\[ \prod_{n=2}^{10} \left(1 - \frac{1}{n^2}\right) = \prod_{n=2}^{10} \frac{(n - 1)(n + 1)}{n^2}. \]

This can be expanded as:

\[ \frac{(1 \cdot 3)(2 \cdot 4)(3 \cdot 5)(4 \cdot 6)(5 \cdot 7)(6 \cdot 8)(7 \cdot 9)(8 \cdot 10)}{(2^2)(3^2)(4^2)(5^2)(6^2)(7^2)(8^2)(9^2)(10^2)}. \]

### Step 3: Write Out the Numerator and Denominator
The numerator consists of the products of pairs:

\[ 1 \cdot 3, \quad 2 \cdot 4, \quad 3 \cdot 5, \quad 4 \cdot 6, \quad 5 \cdot 7, \quad 6 \cdot 8, \quad 7 \cdot 9, \quad 8 \cdot 10. \]

The denominator is:

\[ 2^2 \cdot 3^2 \cdot 4^2 \cdot 5^2 \cdot 6^2 \cdot 7^2 \cdot 8^2 \cdot 9^2 \cdot 10^2. \]

### Step 4: Cancel Common Terms
Notice that in the numerator and denominator, many terms will cancel out. For example, \(2\) from \(2^2\) cancels with \(2\) in the numerator, and so on.

### Step 5: Evaluate the Remaining Terms
After canceling, we will be left with:

\[ \frac{1 \cdot 3 \cdot 2 \cdot 4 \cdot 3 \cdot 5 \cdot 4 \cdot 6 \cdot 5 \cdot 7 \cdot 6 \cdot 8 \cdot 7 \cdot 9 \cdot 8 \cdot 10}{2 \cdot 3 \cdot 4 \cdot 5 \cdot 6 \cdot 7 \cdot 8 \cdot 9 \cdot 10}. \]

### Step 6: Final Calculation
After all cancellations, we find that the product simplifies to:

\[ \frac{11}{2}. \]

Thus, the final value of the expression is:

\[ \frac{11}{2}. \]