Answer :
Certainly! To evaluate the given expression,
[tex]\[ \left(1 - \frac{1}{10}\right)\left(1 - \frac{1}{11}\right)\left(1 - \frac{1}{12}\right) \cdots \left(1 - \frac{1}{99}\right)\left(1 - \frac{1}{100}\right), \][/tex]
we first recognize that each term in the product simplifies to:
[tex]\[ 1 - \frac{1}{n} \][/tex]
where [tex]\( n \)[/tex] ranges from 10 to 100.
Now, let's examine each of these terms individually:
[tex]\[ 1 - \frac{1}{10} = \frac{9}{10}, \][/tex]
[tex]\[ 1 - \frac{1}{11} = \frac{10}{11}, \][/tex]
[tex]\[ 1 - \frac{1}{12} = \frac{11}{12}, \][/tex]
[tex]\[ \quad \vdots \][/tex]
[tex]\[ 1 - \frac{1}{99} = \frac{98}{99}, \][/tex]
[tex]\[ 1 - \frac{1}{100} = \frac{99}{100}. \][/tex]
Putting it all together, our expression becomes:
[tex]\[ \frac{9}{10} \cdot \frac{10}{11} \cdot \frac{11}{12} \cdots \frac{98}{99} \cdot \frac{99}{100}. \][/tex]
Notice that this product is telescoping. Most of the terms in the numerator and denominator cancel out:
[tex]\[ \frac{9 \cdot 10 \cdot 11 \cdots 98 \cdot 99}{10 \cdot 11 \cdot 12 \cdots 99 \cdot 100}. \][/tex]
After cancellation, we are left with:
[tex]\[ \frac{9}{100}. \][/tex]
As a decimal, this is:
[tex]\[ 0.08999999999999997. \][/tex]
Thus, the evaluated product is [tex]\(\boxed{0.09}\)[/tex], which translates to [tex]\(\frac{9}{100}\)[/tex].
Therefore, the correct answer is:
[tex]\[ \boxed{0.09}. \][/tex]
However, please follow the text conversion rule to the correct representation. Given problem options are likely incorrect. Thus, inferred suitable values near [tex]\(\boxed{\frac{9}{100}}\)[/tex] match closest to none given with initial set. Actual valid new approximations pages out [tex]\(\boxed{9/100\)[/tex]} count closest better near $\left(\boxed{1/100}\ `initial set recognized boundary nearness anyway possible writing reality.
[tex]\[ \left(1 - \frac{1}{10}\right)\left(1 - \frac{1}{11}\right)\left(1 - \frac{1}{12}\right) \cdots \left(1 - \frac{1}{99}\right)\left(1 - \frac{1}{100}\right), \][/tex]
we first recognize that each term in the product simplifies to:
[tex]\[ 1 - \frac{1}{n} \][/tex]
where [tex]\( n \)[/tex] ranges from 10 to 100.
Now, let's examine each of these terms individually:
[tex]\[ 1 - \frac{1}{10} = \frac{9}{10}, \][/tex]
[tex]\[ 1 - \frac{1}{11} = \frac{10}{11}, \][/tex]
[tex]\[ 1 - \frac{1}{12} = \frac{11}{12}, \][/tex]
[tex]\[ \quad \vdots \][/tex]
[tex]\[ 1 - \frac{1}{99} = \frac{98}{99}, \][/tex]
[tex]\[ 1 - \frac{1}{100} = \frac{99}{100}. \][/tex]
Putting it all together, our expression becomes:
[tex]\[ \frac{9}{10} \cdot \frac{10}{11} \cdot \frac{11}{12} \cdots \frac{98}{99} \cdot \frac{99}{100}. \][/tex]
Notice that this product is telescoping. Most of the terms in the numerator and denominator cancel out:
[tex]\[ \frac{9 \cdot 10 \cdot 11 \cdots 98 \cdot 99}{10 \cdot 11 \cdot 12 \cdots 99 \cdot 100}. \][/tex]
After cancellation, we are left with:
[tex]\[ \frac{9}{100}. \][/tex]
As a decimal, this is:
[tex]\[ 0.08999999999999997. \][/tex]
Thus, the evaluated product is [tex]\(\boxed{0.09}\)[/tex], which translates to [tex]\(\frac{9}{100}\)[/tex].
Therefore, the correct answer is:
[tex]\[ \boxed{0.09}. \][/tex]
However, please follow the text conversion rule to the correct representation. Given problem options are likely incorrect. Thus, inferred suitable values near [tex]\(\boxed{\frac{9}{100}}\)[/tex] match closest to none given with initial set. Actual valid new approximations pages out [tex]\(\boxed{9/100\)[/tex]} count closest better near $\left(\boxed{1/100}\ `initial set recognized boundary nearness anyway possible writing reality.
