https://file.aiquickdraw.com/mathbot/file/17257065126959lur74t7.jpg

The image shows the following mathematical expression:

\[
\left(1 - \frac{1}{3}\right)\left(1 - \frac{1}{4}\right)\left(1 - \frac{1}{5}\right)\cdots\left(1 - \frac{1}{10}\right)
\]

The available options are:

A) 1  
B) \(\frac{1}{2}\)  
C) \(\frac{1}{4}\)  
D) \(\frac{2}{3}\)

### Solution:

The expression simplifies as follows:

\[
\left(1 - \frac{1}{3}\right) = \frac{2}{3}, \quad \left(1 - \frac{1}{4}\right) = \frac{3}{4}, \quad \left(1 - \frac{1}{5}\right) = \frac{4}{5}, \dots, \left(1 - \frac{1}{10}\right) = \frac{9}{10}
\]

Thus, the product becomes:

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

Notice that most of the terms in the numerator and denominator cancel out:

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

This leaves:

\[
\frac{2}{10} = \frac{1}{5}
\]

Since none of the provided options matches \(\frac{1}{5}\), it seems that the options may have a mistake, or we need to verify further. Based on this product, the correct result of the expression is \(\frac{1}{5}\), but none of the given choices matches this exactly.

Would you like more details or further clarification on any part?

### Related Questions:
1. How would the product look if extended to \( \left(1 - \frac{1}{n}\right) \) for higher \(n\)?
2. What happens if the same approach is used for a different starting value, like \(\left(1 - \frac{1}{2}\right)\)?
3. How can this type of product be generalized for different series?
4. What would happen if each factor had an additional constant added?
5. What is the interpretation of this product in terms of series convergence?

**Tip**: When simplifying long product expressions, always check for cancellation patterns—this often greatly reduces the complexity of the calculations.

Evaluate the expression (1 - 1/3)(1 - 1/4)(1 - 1/5)...(1 - 1/10).

This math problem involves simplifying a product of fractions: (1 - 1/3)(1 - 1/4)(1 - 1/5)...(1 - 1/10). It highlights concepts like multiplying fractions and simplification through cancellation, suitable for students in Grades 7-9.

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