Answer :
To find the value of the given expression:
[tex]\[
\frac{1}{3} \cdot \frac{2}{4} \cdot \frac{3}{5} \cdots \frac{18}{20} \cdot \frac{19}{21} \cdot \frac{20}{22}
\][/tex]
we will follow these steps:
1. Identify the Pattern:
Each fraction in the sequence follows a clear pattern:
- The numerators are [tex]\(1, 2, 3, \ldots, 20\)[/tex].
- The denominators are [tex]\(3, 4, 5, \ldots, 22\)[/tex].
Notice that the denominator is always 2 more than the numerator.
2. Expression as a Product of Fractions:
We can express the entire product as:
[tex]\[
\prod_{i=1}^{20} \frac{i}{i+2}
\][/tex]
This means we will multiply each fraction in the sequence from [tex]\(\frac{1}{3}\)[/tex] to [tex]\(\frac{20}{22}\)[/tex].
3. Simplifying the Product:
A large portion of the terms cancel each other out due to the telescoping nature when expanded:
In detail, when expanded, the sequence looks like this:
[tex]\[
\frac{1 \cdot 2 \cdot 3 \cdot \cdots \cdot 20}{3 \cdot 4 \cdot 5 \cdot \cdots \cdot 22}
\][/tex]
Notice that in the numerator, we have the product [tex]\(1 \times 2 \times 3 \times \cdots \times 20\)[/tex], which can be simply denoted as [tex]\(20!\)[/tex].
Similarly, the denominator is [tex]\(3 \cdot 4 \cdot 5 \cdots \times 22\)[/tex].
4. Ratio and Calculation:
While we know large parts of this sequence telescope (cancel terms), what remains doesn't simplify to integers due to the extending denominators. Therefore, the result of such a product sequence, which doesn't neatly telescope into a simple fraction, approximates to:
[tex]\[
0.004329004329004328
\][/tex]
5. Conclusion:
Therefore, the value of the entire expression is approximately [tex]\(\boxed{0.004329}\)[/tex]. This value is precise and results from calculating the non-cancelled terms of the sequence.
[tex]\[
\frac{1}{3} \cdot \frac{2}{4} \cdot \frac{3}{5} \cdots \frac{18}{20} \cdot \frac{19}{21} \cdot \frac{20}{22}
\][/tex]
we will follow these steps:
1. Identify the Pattern:
Each fraction in the sequence follows a clear pattern:
- The numerators are [tex]\(1, 2, 3, \ldots, 20\)[/tex].
- The denominators are [tex]\(3, 4, 5, \ldots, 22\)[/tex].
Notice that the denominator is always 2 more than the numerator.
2. Expression as a Product of Fractions:
We can express the entire product as:
[tex]\[
\prod_{i=1}^{20} \frac{i}{i+2}
\][/tex]
This means we will multiply each fraction in the sequence from [tex]\(\frac{1}{3}\)[/tex] to [tex]\(\frac{20}{22}\)[/tex].
3. Simplifying the Product:
A large portion of the terms cancel each other out due to the telescoping nature when expanded:
In detail, when expanded, the sequence looks like this:
[tex]\[
\frac{1 \cdot 2 \cdot 3 \cdot \cdots \cdot 20}{3 \cdot 4 \cdot 5 \cdot \cdots \cdot 22}
\][/tex]
Notice that in the numerator, we have the product [tex]\(1 \times 2 \times 3 \times \cdots \times 20\)[/tex], which can be simply denoted as [tex]\(20!\)[/tex].
Similarly, the denominator is [tex]\(3 \cdot 4 \cdot 5 \cdots \times 22\)[/tex].
4. Ratio and Calculation:
While we know large parts of this sequence telescope (cancel terms), what remains doesn't simplify to integers due to the extending denominators. Therefore, the result of such a product sequence, which doesn't neatly telescope into a simple fraction, approximates to:
[tex]\[
0.004329004329004328
\][/tex]
5. Conclusion:
Therefore, the value of the entire expression is approximately [tex]\(\boxed{0.004329}\)[/tex]. This value is precise and results from calculating the non-cancelled terms of the sequence.
